DP Mathematics SL Questionbank
Topic 4 - Vectors
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Description
The aim of this topic is to provide an elementary introduction to vectors, including both algebraic and geometric approaches. The use of dynamic geometry software is extremely helpful to visualize situations in three dimensions.
Directly related questions
- 12M.2.sl.TZ2.8b(i) and (ii): Write down a vector equation for (i) the line OF; (ii) the line AG.
- 08N.1.sl.TZ0.2b: Write down an equation of the line L .
- 08N.2.sl.TZ0.8c(i) and (ii): (i) Find \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} \). (ii) Hence...
- 08M.1.sl.TZ2.8c: The point C is such that...
- 10N.1.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 09M.1.sl.TZ2.10d: The point C is at (2, 1, − 4). Let D be the point such that ABCD is a parallelogram. Find...
- 11N.1.sl.TZ0.8a(i) and (ii): (i) Find \(\overrightarrow {{\rm{PQ}}} \) . (ii) Hence write down a vector equation for...
- 11N.1.sl.TZ0.8b(i) and (ii): (i) Find the value of p . (ii) Given that \({L_2}\) passes through...
- 11N.1.sl.TZ0.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point A. Find the x-coordinate of A.
- 11M.1.sl.TZ1.2b(i) and (ii): Find (i) \(\overrightarrow {{\rm{OP}}} \) ; (ii) \(|\overrightarrow {{\rm{OP}}} |\) .
- 11M.2.sl.TZ2.8c: Find the angle between \({L_1}\) and \({L_2}\) .
- 09N.1.sl.TZ0.2b: Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}} 1 \\ q \\ 5 \end{array}} \right)\)...
- 14M.1.sl.TZ1.8d(i): Write down a direction vector for \({L_2}\).
- 14M.2.sl.TZ1.4b: Given that \(u = \left( \begin{array}{c}3\\2\\1\end{array} \right)\) and...
- 14M.1.sl.TZ2.4c: Write down a vector equation for the line \(L\).
- 09N.2.sl.TZ0.10b: The line \({L_1}\) may be represented by...
- 16N.1.sl.TZ0.8b: Show that the coordinates of C are \(( - 2,{\text{ }}1,{\text{ }}3)\).
- 16M.2.sl.TZ2.10d: Given that...
- 16N.1.sl.TZ0.4a: Find a vector equation of the line that passes through P and Q.
- 17M.1.sl.TZ2.2a: Find the value of \(k\).
- 17M.1.sl.TZ2.9b.ii: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
- 17N.2.sl.TZ0.3b: Let ...
- 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.
- 12N.1.sl.TZ0.6a: Write down a vector equation for line L .
- 08M.1.sl.TZ2.8d: Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) .
- 12M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point R. Find the coordinates of R.
- 10N.1.sl.TZ0.8b(i) and (ii): The line (AC) has equation \({\boldsymbol{r}} = {\boldsymbol{u}} + s{\boldsymbol{v}}\) . (i) ...
- 10N.1.sl.TZ0.8d: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Hence find the...
- 09M.1.sl.TZ2.10a: Write down the equation of \({L_1}\) in the form...
- 09M.1.sl.TZ2.10c: Let B be the point of intersection of lines \({L_1}\) and \({L_2}\) . (i) Show that...
- 10N.2.sl.TZ0.4: Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}}2\\{ - 3}\\6\end{array}} \right)\) and ...
- SPNone.1.sl.TZ0.1a: \(\overrightarrow {{\rm{AE}}} \)
- SPNone.2.sl.TZ0.4b: Write down the position vector of the point of intersection of \({L_1}\) and \({L_2}\) .
- 13M.1.sl.TZ1.8b: Given that \({L_1}\) is perpendicular to \({L_2}\) , show that \(p = - 6\) .
- 14M.1.sl.TZ1.8c: Find the coordinates of P.
- 14M.1.sl.TZ1.8d(ii): Hence, find the angle between \({L_1}\) and \({L_2}\).
- 14N.1.sl.TZ0.10a: Write down the equation of \({L_1}\).
- 15M.1.sl.TZ1.8a: (i) Show that...
- 15M.2.sl.TZ2.2b: Find the angle between \({{u}}\) and \({{v}}\).
- 16N.1.sl.TZ0.8d: Given that...
- 17M.1.sl.TZ1.8d.ii: Hence or otherwise, find one point on \({L_2}\) which is \(\sqrt 5 \) units from C.
- 17N.2.sl.TZ0.3a: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
- 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
- 12N.1.sl.TZ0.6b: The line L intersects the x-axis at the point P. Find the x-coordinate of P.
- 12M.2.sl.TZ2.8c: Find the obtuse angle between the lines OF and AG.
- 08M.2.sl.TZ1.9a(i) and (ii): (i) Show that...
- 08M.2.sl.TZ1.9d: The line \({L_3}\) has equation...
- 09M.1.sl.TZ2.10b: The line \({L_2}\) has equation...
- 13M.2.sl.TZ2.8a: Find (i) \(\overrightarrow {{\rm{AB}}} \) ; (ii) \(\overrightarrow {{\rm{AC}}} \) .
- 13M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\) .
- 14M.1.sl.TZ1.8b(ii): Hence, write down a vector equation for \({L_1}\).
- 15N.1.sl.TZ0.9b: A line \({L_2}\) has equation...
- 15N.1.sl.TZ0.9c: A point \(D\) lies on line \({L_2}\) so that...
- 16M.2.sl.TZ1.10c: Find \(\theta \).
- 16N.1.sl.TZ0.4b: The line through P and Q is perpendicular to the vector 2i \( + \) nk. Find the value of \(n\).
- 17M.1.sl.TZ1.8b: A second line \({L_2}\), has equation r =...
- 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
- 12M.2.sl.TZ2.8a(i), (ii) and (iii): (i) Find \(\overrightarrow {{\rm{OB}}} \) . (ii) Find \(\overrightarrow {{\rm{OF}}} \)...
- 08M.1.sl.TZ2.8a(i) and (ii): Find (i) \(\overrightarrow {{\rm{AB}}} \) ; (ii) \(\overrightarrow {{\rm{AD}}} \) giving...
- 12M.1.sl.TZ1.8a(i) and (ii): (i) Show that...
- 10M.1.sl.TZ2.2b: Find a unit vector in the direction of \(\overrightarrow {{\rm{AB}}} \) .
- 10M.1.sl.TZ2.2a: Find \(\overrightarrow {{\rm{BC}}} \) .
- 09N.2.sl.TZ0.10a: Show...
- 09N.2.sl.TZ0.10c: The point \({\text{T}}( - 1{\text{, }}5{\text{, }}p)\) lies on \({L_1}\) . Find the value of...
- SPNone.1.sl.TZ0.1b: \(\overrightarrow {{\rm{EC}}} \)
- SPNone.2.sl.TZ0.4c: Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of a .
- 13N.2.sl.TZ0.9a: Find the coordinates of \({\text{P}}\).
- 13M.1.sl.TZ1.1b: Find \(\boldsymbol{c}\) .
- 15M.1.sl.TZ1.8c: The following diagram shows the line \(L\) and the origin \(O\). The point \(C\) also lies on...
- 16N.1.sl.TZ0.8a: (i) Find \(\overrightarrow {{\text{AB}}} \). (ii) Find...
- 16M.2.sl.TZ2.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 17M.1.sl.TZ1.8d.i: Find a unit vector in the direction of \({L_2}\).
- 17M.1.sl.TZ2.2b: Given that c = a + 2b, find c.
- 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
- 18M.1.sl.TZ2.1b: The vector \(\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)\)...
- 12N.1.sl.TZ0.9b: Let C and D be points such that ABCD is a rectangle. Given that...
- 08N.2.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 08M.2.sl.TZ1.7: Let \({\boldsymbol{v}} = 3{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) and...
- 08M.2.sl.TZ1.9b: The line \({L_1}\) has...
- 08M.1.sl.TZ2.8b: Show that \(k = 7\) .
- 12M.1.sl.TZ1.8b: Find the cosine of the angle between \(\overrightarrow {{\rm{PQ}}} \) and \({L_2}\) .
- 10N.1.sl.TZ0.8c: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Show that...
- 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
- 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
- 09N.2.sl.TZ0.10e: Let \(\theta \) be the obtuse angle between \({L_1}\) and \({L_2}\) . Calculate the size of...
- 09M.1.sl.TZ1.9b: Show that \(\cos {\rm{R}}\widehat {\rm{P}}{\rm{Q}} = \frac{1}{2}\) .
- 09M.2.sl.TZ1.5: Two lines with equations...
- 09M.1.sl.TZ2.2: Find the cosine of the angle between the two vectors...
- 11M.1.sl.TZ1.9a(i) and (ii): (i) Write down \(\overrightarrow {{\rm{BA}}} \) . (ii) Find...
- 11M.1.sl.TZ2.3c: Show that vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are...
- 11M.2.sl.TZ2.8a: Find \(\overrightarrow {{\rm{AB}}} \) .
- 13M.2.sl.TZ2.8c: i. Show that \(\cos q = \frac{{2a + 14}}{{\sqrt {14{a^2} + 280} }}\) . ii. Hence, find the value...
- 14M.1.sl.TZ1.8a: Show that \(\overrightarrow {{\text{AB}}} =...
- 14M.1.sl.TZ2.4a: Find the gradient of the line \(L\).
- 14M.1.sl.TZ2.9d: The two airplanes collide at the point \((-23, 20, 28)\). How long after Ryan’s airplane takes...
- 14N.1.sl.TZ0.7: The following diagram shows triangle \(ABC\). Let...
- 16M.2.sl.TZ2.10e: The point D lies on \(L\) such that...
- 17M.1.sl.TZ2.9c: Find \(\cos {\rm{B\hat AC}}\).
- 17N.1.sl.TZ0.9a.i: Show that...
- 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
- 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
- 12N.1.sl.TZ0.9a: Show that...
- 08N.2.sl.TZ0.8b: Find the coordinates of point C.
- 09N.1.sl.TZ0.2a: Let u \( = \left( {\begin{array}{*{20}{c}}2\\3\\{ - 1}\end{array}} \right)\) and w...
- 09M.1.sl.TZ1.9a: Find (i) \(\overrightarrow {{\rm{PQ}}} \) ; (ii) \(\overrightarrow {{\rm{PR}}} \) .
- 11M.1.sl.TZ2.3a: Find \(\overrightarrow {{\rm{BC}}} \) .
- 13M.1.sl.TZ1.8c: The line \({L_1}\) intersects the line \({L_2}\) at point Q. Find the \(x\)-coordinate of Q.
- 14M.1.sl.TZ2.9b: Find the height of Ryan’s airplane after two seconds.
- 13N.2.sl.TZ0.9c: The point \({\text{Q}}(7, 5, 3)\) lies on \({L_1}\). The point \({\text{R}}\) is the reflection...
- 13N.2.sl.TZ0.9b: Show that the lines are perpendicular.
- 13M.1.sl.TZ1.1a: Find (i) \(2\boldsymbol{a} + \boldsymbol{b}\) ; (ii) ...
- 14N.1.sl.TZ0.10b: A line \({L_a}\) crosses the \(y\)-axis at a point \(P\). Show that \(P\) has coordinates...
- 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
- 16M.2.sl.TZ1.10b: Show that...
- 16M.2.sl.TZ1.10d: (i) Show that...
- 16M.2.sl.TZ2.10b: Find the coordinates of C.
- 17M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_1}\) intersect at \(C(9,{\text{ }}13,{\text{ }}z)\). Find \(z\).
- 17M.1.sl.TZ2.9a: Find the coordinates of A.
- 17M.1.sl.TZ2.9d: Hence or otherwise, find the distance between \({P_1}\) and \({P_2}\) two seconds after they...
- 17N.1.sl.TZ0.9a.ii: Find a vector equation for \(L\).
- 17N.1.sl.TZ0.9b: Find the value of \(p\).
- 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to \).
- 18M.1.sl.TZ2.1a: Find a vector equation for L1.
- 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
- 08N.1.sl.TZ0.2a(i) and (ii): (i) Find the velocity vector, \(\overrightarrow {{\rm{AB}}} \) . (ii) Find the speed of...
- 08M.2.sl.TZ2.7: The line L1 is represented by...
- 10M.1.sl.TZ2.2c: Show that \(\overrightarrow {{\rm{AB}}} \) is perpendicular to \(\overrightarrow {{\rm{AC}}} \) .
- 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
- SPNone.2.sl.TZ0.4a: Write down the line that is parallel to \({L_4}\) .
- 11M.1.sl.TZ1.2a: Write down a vector equation for L in the form...
- 11M.1.sl.TZ2.3b: Show...
- 13M.1.sl.TZ1.8a.i: Find \(\overrightarrow {{\rm{AB}}} \) .
- 13M.2.sl.TZ2.8b: Find the value of \(a\) for which \({\rm{q}} = \frac{\pi }{2}\) .
- 14M.1.sl.TZ2.9c: The position of Jack’s airplane \(s\) seconds after it takes off is given by r =...
- 13N.1.sl.TZ0.1b: \(\overrightarrow {{\text{OT}}} \).
- 15M.1.sl.TZ1.8d: (i) Find \(\overrightarrow {{\text{OC}}} \bullet \overrightarrow {{\text{AB}}} \). (ii) ...
- 15M.1.sl.TZ1.8e: Hence or otherwise, find the area of triangle \(OAB\).
- 15M.2.sl.TZ2.2a: Find (i) \(u \bullet v\); (ii) \(\left| {{u}} \right|\); (iii) ...
- 17M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\).
- 17N.1.sl.TZ0.9c: The point D has coordinates \(({q^2},{\text{ }}0,{\text{ }}q)\). Given that...
- 18M.1.sl.TZ1.9a: Show...
- 12N.1.sl.TZ0.9c: Let C and D be points such that ABCD is a rectangle. Find the coordinates of point C.
- 12N.1.sl.TZ0.9d: Let C and D be points such that ABCD is a rectangle. Find the area of rectangle ABCD.
- 08M.2.sl.TZ1.9c(i) and (ii): The line \({L_2}\) passes through A and is parallel to \(\overrightarrow {{\rm{OB}}} \) . (i) ...
- 09N.2.sl.TZ0.10d: The point T also lies on \({L_2}\) with equation...
- 10M.2.sl.TZ2.9a(i) and (ii): (i) Write down the coordinates of A. (ii) Find the speed of the airplane in...
- 10M.2.sl.TZ2.9b(i) and (ii): After seven seconds the airplane passes through a point B. (i) Find the coordinates of...
- 10M.2.sl.TZ2.9c: Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by...
- 11M.1.sl.TZ1.9b(i) and (ii): (i) Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) . (ii) Hence, find...
- 11M.1.sl.TZ1.9c(i) and (ii): The point D is such that...
- 11M.2.sl.TZ2.8d: The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.
- 11M.2.sl.TZ2.8b: Find an equation for \({L_1}\) in the form...
- 13M.2.sl.TZ2.4: Line \({L_1}\) has...
- 14M.1.sl.TZ1.8b(i): Hence, write down a direction vector for \({L_1}\);
- 14M.1.sl.TZ2.9a: Find the speed of Ryan’s airplane.
- 13N.1.sl.TZ0.1a: \(\overrightarrow {{\text{QP}}} \);
- 14M.2.sl.TZ1.4a: Let \(w = u - v\). Represent \(w\) on the diagram above.
- 16M.2.sl.TZ1.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 16M.1.sl.TZ2.7: Let u \( = - 3\)i \( + \) j \( + \) k and v \( = m\)j \( + {\text{ }}n\)k , where...
- 16M.2.sl.TZ2.10c: Write down a vector equation for \(L\).
- 17M.1.sl.TZ2.9b.i: Find \(\overrightarrow {{\text{AB}}} \);
- 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
- 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to \).
- 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).
Sub sections and their related questions
4.1
- 12N.1.sl.TZ0.9a: Show that...
- 12N.1.sl.TZ0.9b: Let C and D be points such that ABCD is a rectangle. Given that...
- 12N.1.sl.TZ0.9c: Let C and D be points such that ABCD is a rectangle. Find the coordinates of point C.
- 12N.1.sl.TZ0.9d: Let C and D be points such that ABCD is a rectangle. Find the area of rectangle ABCD.
- 12M.2.sl.TZ2.8a(i), (ii) and (iii): (i) Find \(\overrightarrow {{\rm{OB}}} \) . (ii) Find \(\overrightarrow {{\rm{OF}}} \)...
- 12M.2.sl.TZ2.8b(i) and (ii): Write down a vector equation for (i) the line OF; (ii) the line AG.
- 12M.2.sl.TZ2.8c: Find the obtuse angle between the lines OF and AG.
- 08N.1.sl.TZ0.2a(i) and (ii): (i) Find the velocity vector, \(\overrightarrow {{\rm{AB}}} \) . (ii) Find the speed of...
- 08N.2.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 08N.2.sl.TZ0.8b: Find the coordinates of point C.
- 08M.2.sl.TZ1.7: Let \({\boldsymbol{v}} = 3{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) and...
- 08M.2.sl.TZ1.9a(i) and (ii): (i) Show that...
- 08M.1.sl.TZ2.8a(i) and (ii): Find (i) \(\overrightarrow {{\rm{AB}}} \) ; (ii) \(\overrightarrow {{\rm{AD}}} \) giving...
- 08M.1.sl.TZ2.8c: The point C is such that...
- 12M.1.sl.TZ1.8a(i) and (ii): (i) Show that...
- 12M.1.sl.TZ1.8b: Find the cosine of the angle between \(\overrightarrow {{\rm{PQ}}} \) and \({L_2}\) .
- 12M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point R. Find the coordinates of R.
- 10N.1.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 10N.1.sl.TZ0.8b(i) and (ii): The line (AC) has equation \({\boldsymbol{r}} = {\boldsymbol{u}} + s{\boldsymbol{v}}\) . (i) ...
- 10N.1.sl.TZ0.8c: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Show that...
- 10N.1.sl.TZ0.8d: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Hence find the...
- 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
- 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
- 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
- 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
- 10M.1.sl.TZ2.2a: Find \(\overrightarrow {{\rm{BC}}} \) .
- 10M.1.sl.TZ2.2b: Find a unit vector in the direction of \(\overrightarrow {{\rm{AB}}} \) .
- 10M.1.sl.TZ2.2c: Show that \(\overrightarrow {{\rm{AB}}} \) is perpendicular to \(\overrightarrow {{\rm{AC}}} \) .
- 09N.2.sl.TZ0.10a: Show...
- 09M.1.sl.TZ1.9a: Find (i) \(\overrightarrow {{\rm{PQ}}} \) ; (ii) \(\overrightarrow {{\rm{PR}}} \) .
- 09M.1.sl.TZ2.10c: Let B be the point of intersection of lines \({L_1}\) and \({L_2}\) . (i) Show that...
- 09M.1.sl.TZ2.10d: The point C is at (2, 1, − 4). Let D be the point such that ABCD is a parallelogram. Find...
- 10M.2.sl.TZ2.9a(i) and (ii): (i) Write down the coordinates of A. (ii) Find the speed of the airplane in...
- 10M.2.sl.TZ2.9b(i) and (ii): After seven seconds the airplane passes through a point B. (i) Find the coordinates of...
- 10M.2.sl.TZ2.9c: Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by...
- SPNone.1.sl.TZ0.1a: \(\overrightarrow {{\rm{AE}}} \)
- SPNone.1.sl.TZ0.1b: \(\overrightarrow {{\rm{EC}}} \)
- 11N.1.sl.TZ0.8a(i) and (ii): (i) Find \(\overrightarrow {{\rm{PQ}}} \) . (ii) Hence write down a vector equation for...
- 11N.1.sl.TZ0.8b(i) and (ii): (i) Find the value of p . (ii) Given that \({L_2}\) passes through...
- 11N.1.sl.TZ0.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point A. Find the x-coordinate of A.
- 11M.1.sl.TZ1.2a: Write down a vector equation for L in the form...
- 11M.1.sl.TZ1.2b(i) and (ii): Find (i) \(\overrightarrow {{\rm{OP}}} \) ; (ii) \(|\overrightarrow {{\rm{OP}}} |\) .
- 11M.1.sl.TZ1.9a(i) and (ii): (i) Write down \(\overrightarrow {{\rm{BA}}} \) . (ii) Find...
- 11M.1.sl.TZ1.9b(i) and (ii): (i) Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) . (ii) Hence, find...
- 11M.1.sl.TZ1.9c(i) and (ii): The point D is such that...
- 11M.1.sl.TZ2.3a: Find \(\overrightarrow {{\rm{BC}}} \) .
- 11M.1.sl.TZ2.3b: Show...
- 11M.1.sl.TZ2.3c: Show that vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are...
- 11M.2.sl.TZ2.8a: Find \(\overrightarrow {{\rm{AB}}} \) .
- 11M.2.sl.TZ2.8b: Find an equation for \({L_1}\) in the form...
- 11M.2.sl.TZ2.8c: Find the angle between \({L_1}\) and \({L_2}\) .
- 11M.2.sl.TZ2.8d: The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.
- 13M.1.sl.TZ1.8a.i: Find \(\overrightarrow {{\rm{AB}}} \) .
- 13M.2.sl.TZ2.8a: Find (i) \(\overrightarrow {{\rm{AB}}} \) ; (ii) \(\overrightarrow {{\rm{AC}}} \) .
- 09N.1.sl.TZ0.2b: Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}} 1 \\ q \\ 5 \end{array}} \right)\)...
- 14M.1.sl.TZ1.8a: Show that \(\overrightarrow {{\text{AB}}} =...
- 13N.1.sl.TZ0.1a: \(\overrightarrow {{\text{QP}}} \);
- 13N.1.sl.TZ0.1b: \(\overrightarrow {{\text{OT}}} \).
- 13N.2.sl.TZ0.9c: The point \({\text{Q}}(7, 5, 3)\) lies on \({L_1}\). The point \({\text{R}}\) is the reflection...
- 14M.2.sl.TZ1.4a: Let \(w = u - v\). Represent \(w\) on the diagram above.
- 13M.1.sl.TZ1.1a: Find (i) \(2\boldsymbol{a} + \boldsymbol{b}\) ; (ii) ...
- 13M.1.sl.TZ1.1b: Find \(\boldsymbol{c}\) .
- 15M.1.sl.TZ1.8a: (i) Show that...
- 15M.1.sl.TZ1.8e: Hence or otherwise, find the area of triangle \(OAB\).
- 15M.2.sl.TZ2.2a: Find (i) \(u \bullet v\); (ii) \(\left| {{u}} \right|\); (iii) ...
- 15N.1.sl.TZ0.9c: A point \(D\) lies on line \({L_2}\) so that...
- 16M.2.sl.TZ1.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 16M.2.sl.TZ1.10b: Show that...
- 16M.2.sl.TZ1.10c: Find \(\theta \).
- 16M.2.sl.TZ1.10d: (i) Show that...
- 16M.1.sl.TZ2.7: Let u \( = - 3\)i \( + \) j \( + \) k and v \( = m\)j \( + {\text{ }}n\)k , where...
- 16M.2.sl.TZ2.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 16M.2.sl.TZ2.10b: Find the coordinates of C.
- 16M.2.sl.TZ2.10c: Write down a vector equation for \(L\).
- 16M.2.sl.TZ2.10d: Given that...
- 16M.2.sl.TZ2.10e: The point D lies on \(L\) such that...
- 16N.1.sl.TZ0.8a: (i) Find \(\overrightarrow {{\text{AB}}} \). (ii) Find...
- 16N.1.sl.TZ0.8b: Show that the coordinates of C are \(( - 2,{\text{ }}1,{\text{ }}3)\).
- 16N.1.sl.TZ0.8d: Given that...
- 17M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\).
- 17M.1.sl.TZ1.8b: A second line \({L_2}\), has equation r =...
- 17M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_1}\) intersect at \(C(9,{\text{ }}13,{\text{ }}z)\). Find \(z\).
- 17M.1.sl.TZ1.8d.i: Find a unit vector in the direction of \({L_2}\).
- 17M.1.sl.TZ1.8d.ii: Hence or otherwise, find one point on \({L_2}\) which is \(\sqrt 5 \) units from C.
- 17M.1.sl.TZ2.2a: Find the value of \(k\).
- 17M.1.sl.TZ2.2b: Given that c = a + 2b, find c.
- 17M.1.sl.TZ2.9a: Find the coordinates of A.
- 17M.1.sl.TZ2.9b.i: Find \(\overrightarrow {{\text{AB}}} \);
- 17M.1.sl.TZ2.9c: Find \(\cos {\rm{B\hat AC}}\).
- 17M.1.sl.TZ2.9d: Hence or otherwise, find the distance between \({P_1}\) and \({P_2}\) two seconds after they...
- 17M.1.sl.TZ2.9b.ii: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
- 17N.1.sl.TZ0.9a.i: Show that...
- 17N.1.sl.TZ0.9a.ii: Find a vector equation for \(L\).
- 17N.1.sl.TZ0.9b: Find the value of \(p\).
- 17N.1.sl.TZ0.9c: The point D has coordinates \(({q^2},{\text{ }}0,{\text{ }}q)\). Given that...
- 17N.2.sl.TZ0.3a: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
- 17N.2.sl.TZ0.3b: Let ...
- 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
- 18M.1.sl.TZ1.9a: Show...
- 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
- 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
- 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to \).
- 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
- 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
- 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to \).
- 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).
- 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
- 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
- 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.
4.2
- 12N.1.sl.TZ0.9a: Show that...
- 12N.1.sl.TZ0.9b: Let C and D be points such that ABCD is a rectangle. Given that...
- 12N.1.sl.TZ0.9c: Let C and D be points such that ABCD is a rectangle. Find the coordinates of point C.
- 12N.1.sl.TZ0.9d: Let C and D be points such that ABCD is a rectangle. Find the area of rectangle ABCD.
- 08N.2.sl.TZ0.8c(i) and (ii): (i) Find \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} \). (ii) Hence...
- 08M.2.sl.TZ1.7: Let \({\boldsymbol{v}} = 3{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) and...
- 08M.2.sl.TZ1.9a(i) and (ii): (i) Show that...
- 08M.1.sl.TZ2.8b: Show that \(k = 7\) .
- 08M.1.sl.TZ2.8d: Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) .
- 10N.1.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 10N.1.sl.TZ0.8b(i) and (ii): The line (AC) has equation \({\boldsymbol{r}} = {\boldsymbol{u}} + s{\boldsymbol{v}}\) . (i) ...
- 10N.1.sl.TZ0.8c: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Show that...
- 10N.1.sl.TZ0.8d: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Hence find the...
- 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
- 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
- 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
- 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
- 10M.1.sl.TZ2.2a: Find \(\overrightarrow {{\rm{BC}}} \) .
- 10M.1.sl.TZ2.2b: Find a unit vector in the direction of \(\overrightarrow {{\rm{AB}}} \) .
- 10M.1.sl.TZ2.2c: Show that \(\overrightarrow {{\rm{AB}}} \) is perpendicular to \(\overrightarrow {{\rm{AC}}} \) .
- 09N.1.sl.TZ0.2a: Let u \( = \left( {\begin{array}{*{20}{c}}2\\3\\{ - 1}\end{array}} \right)\) and w...
- 09N.2.sl.TZ0.10e: Let \(\theta \) be the obtuse angle between \({L_1}\) and \({L_2}\) . Calculate the size of...
- 09M.1.sl.TZ1.9b: Show that \(\cos {\rm{R}}\widehat {\rm{P}}{\rm{Q}} = \frac{1}{2}\) .
- 09M.1.sl.TZ2.2: Find the cosine of the angle between the two vectors...
- 10N.2.sl.TZ0.4: Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}}2\\{ - 3}\\6\end{array}} \right)\) and ...
- SPNone.2.sl.TZ0.4a: Write down the line that is parallel to \({L_4}\) .
- SPNone.2.sl.TZ0.4c: Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of a .
- 11N.1.sl.TZ0.8a(i) and (ii): (i) Find \(\overrightarrow {{\rm{PQ}}} \) . (ii) Hence write down a vector equation for...
- 11N.1.sl.TZ0.8b(i) and (ii): (i) Find the value of p . (ii) Given that \({L_2}\) passes through...
- 11N.1.sl.TZ0.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point A. Find the x-coordinate of A.
- 11M.1.sl.TZ1.9a(i) and (ii): (i) Write down \(\overrightarrow {{\rm{BA}}} \) . (ii) Find...
- 11M.1.sl.TZ1.9b(i) and (ii): (i) Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) . (ii) Hence, find...
- 11M.1.sl.TZ1.9c(i) and (ii): The point D is such that...
- 11M.1.sl.TZ2.3a: Find \(\overrightarrow {{\rm{BC}}} \) .
- 11M.1.sl.TZ2.3b: Show...
- 11M.1.sl.TZ2.3c: Show that vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are...
- 13M.1.sl.TZ1.8b: Given that \({L_1}\) is perpendicular to \({L_2}\) , show that \(p = - 6\) .
- 13M.2.sl.TZ2.8b: Find the value of \(a\) for which \({\rm{q}} = \frac{\pi }{2}\) .
- 13M.2.sl.TZ2.8c: i. Show that \(\cos q = \frac{{2a + 14}}{{\sqrt {14{a^2} + 280} }}\) . ii. Hence, find the value...
- 14M.2.sl.TZ1.4b: Given that \(u = \left( \begin{array}{c}3\\2\\1\end{array} \right)\) and...
- 14M.1.sl.TZ2.4a: Find the gradient of the line \(L\).
- 14M.1.sl.TZ2.9c: The position of Jack’s airplane \(s\) seconds after it takes off is given by r =...
- 13N.2.sl.TZ0.9b: Show that the lines are perpendicular.
- 14N.1.sl.TZ0.7: The following diagram shows triangle \(ABC\). Let...
- 15M.1.sl.TZ1.8d: (i) Find \(\overrightarrow {{\text{OC}}} \bullet \overrightarrow {{\text{AB}}} \). (ii) ...
- 15M.2.sl.TZ2.2a: Find (i) \(u \bullet v\); (ii) \(\left| {{u}} \right|\); (iii) ...
- 15M.2.sl.TZ2.2b: Find the angle between \({{u}}\) and \({{v}}\).
- 15N.1.sl.TZ0.9c: A point \(D\) lies on line \({L_2}\) so that...
- 16M.2.sl.TZ1.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 16M.2.sl.TZ1.10b: Show that...
- 16M.2.sl.TZ1.10c: Find \(\theta \).
- 16M.2.sl.TZ1.10d: (i) Show that...
- 16M.1.sl.TZ2.7: Let u \( = - 3\)i \( + \) j \( + \) k and v \( = m\)j \( + {\text{ }}n\)k , where...
- 16N.1.sl.TZ0.4b: The line through P and Q is perpendicular to the vector 2i \( + \) nk. Find the value of \(n\).
- 17M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\).
- 17M.1.sl.TZ1.8b: A second line \({L_2}\), has equation r =...
- 17M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_1}\) intersect at \(C(9,{\text{ }}13,{\text{ }}z)\). Find \(z\).
- 17M.1.sl.TZ1.8d.i: Find a unit vector in the direction of \({L_2}\).
- 17M.1.sl.TZ1.8d.ii: Hence or otherwise, find one point on \({L_2}\) which is \(\sqrt 5 \) units from C.
- 17M.1.sl.TZ2.2a: Find the value of \(k\).
- 17M.1.sl.TZ2.2b: Given that c = a + 2b, find c.
- 17N.1.sl.TZ0.9a.i: Show that...
- 17N.1.sl.TZ0.9a.ii: Find a vector equation for \(L\).
- 17N.1.sl.TZ0.9b: Find the value of \(p\).
- 17N.1.sl.TZ0.9c: The point D has coordinates \(({q^2},{\text{ }}0,{\text{ }}q)\). Given that...
- 17N.2.sl.TZ0.3a: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
- 17N.2.sl.TZ0.3b: Let ...
- 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
- 18M.1.sl.TZ1.9a: Show...
- 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
- 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
- 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to \).
- 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
- 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
- 18M.1.sl.TZ2.1a: Find a vector equation for L1.
- 18M.1.sl.TZ2.1b: The vector \(\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)\)...
- 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to \).
- 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).
- 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
- 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
- 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.
4.3
- 12N.1.sl.TZ0.6a: Write down a vector equation for line L .
- 12N.1.sl.TZ0.6b: The line L intersects the x-axis at the point P. Find the x-coordinate of P.
- 12M.2.sl.TZ2.8a(i), (ii) and (iii): (i) Find \(\overrightarrow {{\rm{OB}}} \) . (ii) Find \(\overrightarrow {{\rm{OF}}} \)...
- 12M.2.sl.TZ2.8b(i) and (ii): Write down a vector equation for (i) the line OF; (ii) the line AG.
- 12M.2.sl.TZ2.8c: Find the obtuse angle between the lines OF and AG.
- 08N.1.sl.TZ0.2a(i) and (ii): (i) Find the velocity vector, \(\overrightarrow {{\rm{AB}}} \) . (ii) Find the speed of...
- 08N.1.sl.TZ0.2b: Write down an equation of the line L .
- 08M.2.sl.TZ1.9b: The line \({L_1}\) has...
- 08M.2.sl.TZ1.9c(i) and (ii): The line \({L_2}\) passes through A and is parallel to \(\overrightarrow {{\rm{OB}}} \) . (i) ...
- 08M.2.sl.TZ1.9d: The line \({L_3}\) has equation...
- 12M.1.sl.TZ1.8a(i) and (ii): (i) Show that...
- 12M.1.sl.TZ1.8b: Find the cosine of the angle between \(\overrightarrow {{\rm{PQ}}} \) and \({L_2}\) .
- 12M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point R. Find the coordinates of R.
- 10N.1.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 10N.1.sl.TZ0.8b(i) and (ii): The line (AC) has equation \({\boldsymbol{r}} = {\boldsymbol{u}} + s{\boldsymbol{v}}\) . (i) ...
- 10N.1.sl.TZ0.8c: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Show that...
- 10N.1.sl.TZ0.8d: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Hence find the...
- 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
- 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
- 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
- 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
- 09N.2.sl.TZ0.10b: The line \({L_1}\) may be represented by...
- 09N.2.sl.TZ0.10c: The point \({\text{T}}( - 1{\text{, }}5{\text{, }}p)\) lies on \({L_1}\) . Find the value of...
- 09N.2.sl.TZ0.10d: The point T also lies on \({L_2}\) with equation...
- 09M.1.sl.TZ2.10a: Write down the equation of \({L_1}\) in the form...
- 09M.1.sl.TZ2.10b: The line \({L_2}\) has equation...
- 10M.2.sl.TZ2.9a(i) and (ii): (i) Write down the coordinates of A. (ii) Find the speed of the airplane in...
- 10M.2.sl.TZ2.9b(i) and (ii): After seven seconds the airplane passes through a point B. (i) Find the coordinates of...
- 10M.2.sl.TZ2.9c: Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by...
- SPNone.2.sl.TZ0.4a: Write down the line that is parallel to \({L_4}\) .
- SPNone.2.sl.TZ0.4b: Write down the position vector of the point of intersection of \({L_1}\) and \({L_2}\) .
- SPNone.2.sl.TZ0.4c: Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of a .
- 11N.1.sl.TZ0.8a(i) and (ii): (i) Find \(\overrightarrow {{\rm{PQ}}} \) . (ii) Hence write down a vector equation for...
- 11N.1.sl.TZ0.8b(i) and (ii): (i) Find the value of p . (ii) Given that \({L_2}\) passes through...
- 11N.1.sl.TZ0.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point A. Find the x-coordinate of A.
- 11M.1.sl.TZ1.2a: Write down a vector equation for L in the form...
- 11M.1.sl.TZ1.2b(i) and (ii): Find (i) \(\overrightarrow {{\rm{OP}}} \) ; (ii) \(|\overrightarrow {{\rm{OP}}} |\) .
- 11M.2.sl.TZ2.8a: Find \(\overrightarrow {{\rm{AB}}} \) .
- 11M.2.sl.TZ2.8b: Find an equation for \({L_1}\) in the form...
- 11M.2.sl.TZ2.8c: Find the angle between \({L_1}\) and \({L_2}\) .
- 11M.2.sl.TZ2.8d: The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.
- 13M.1.sl.TZ1.8b: Given that \({L_1}\) is perpendicular to \({L_2}\) , show that \(p = - 6\) .
- 13M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\) .
- 14M.1.sl.TZ1.8b(i): Hence, write down a direction vector for \({L_1}\);
- 14M.1.sl.TZ1.8b(ii): Hence, write down a vector equation for \({L_1}\).
- 14M.1.sl.TZ1.8d(i): Write down a direction vector for \({L_2}\).
- 14M.1.sl.TZ1.8d(ii): Hence, find the angle between \({L_1}\) and \({L_2}\).
- 14M.1.sl.TZ2.4c: Write down a vector equation for the line \(L\).
- 14M.1.sl.TZ2.9a: Find the speed of Ryan’s airplane.
- 14M.1.sl.TZ2.9b: Find the height of Ryan’s airplane after two seconds.
- 14N.1.sl.TZ0.10a: Write down the equation of \({L_1}\).
- 14N.1.sl.TZ0.10b: A line \({L_a}\) crosses the \(y\)-axis at a point \(P\). Show that \(P\) has coordinates...
- 15M.1.sl.TZ1.8c: The following diagram shows the line \(L\) and the origin \(O\). The point \(C\) also lies on...
- 16M.2.sl.TZ1.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 16M.2.sl.TZ1.10b: Show that...
- 16M.2.sl.TZ1.10c: Find \(\theta \).
- 16M.2.sl.TZ1.10d: (i) Show that...
- 16M.2.sl.TZ2.10a: Find \(\overrightarrow {{\text{AB}}} \).
- 16M.2.sl.TZ2.10b: Find the coordinates of C.
- 16M.2.sl.TZ2.10c: Write down a vector equation for \(L\).
- 16M.2.sl.TZ2.10d: Given that...
- 16M.2.sl.TZ2.10e: The point D lies on \(L\) such that...
- 16N.1.sl.TZ0.4a: Find a vector equation of the line that passes through P and Q.
- 17M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\).
- 17M.1.sl.TZ1.8b: A second line \({L_2}\), has equation r =...
- 17M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_1}\) intersect at \(C(9,{\text{ }}13,{\text{ }}z)\). Find \(z\).
- 17M.1.sl.TZ1.8d.i: Find a unit vector in the direction of \({L_2}\).
- 17M.1.sl.TZ1.8d.ii: Hence or otherwise, find one point on \({L_2}\) which is \(\sqrt 5 \) units from C.
- 17M.1.sl.TZ2.9a: Find the coordinates of A.
- 17M.1.sl.TZ2.9b.i: Find \(\overrightarrow {{\text{AB}}} \);
- 17M.1.sl.TZ2.9c: Find \(\cos {\rm{B\hat AC}}\).
- 17M.1.sl.TZ2.9d: Hence or otherwise, find the distance between \({P_1}\) and \({P_2}\) two seconds after they...
- 17M.1.sl.TZ2.9b.ii: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
- 17N.1.sl.TZ0.9a.i: Show that...
- 17N.1.sl.TZ0.9a.ii: Find a vector equation for \(L\).
- 17N.1.sl.TZ0.9b: Find the value of \(p\).
- 17N.1.sl.TZ0.9c: The point D has coordinates \(({q^2},{\text{ }}0,{\text{ }}q)\). Given that...
- 18M.1.sl.TZ1.9a: Show...
- 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
- 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
- 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to \).
- 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
- 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
- 18M.1.sl.TZ2.1a: Find a vector equation for L1.
- 18M.1.sl.TZ2.1b: The vector \(\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)\)...
4.4
- 12N.1.sl.TZ0.6a: Write down a vector equation for line L .
- 12N.1.sl.TZ0.6b: The line L intersects the x-axis at the point P. Find the x-coordinate of P.
- 08M.2.sl.TZ2.7: The line L1 is represented by...
- 12M.1.sl.TZ1.8a(i) and (ii): (i) Show that...
- 12M.1.sl.TZ1.8b: Find the cosine of the angle between \(\overrightarrow {{\rm{PQ}}} \) and \({L_2}\) .
- 12M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point R. Find the coordinates of R.
- 10N.1.sl.TZ0.8a(i), (ii) and (iii): (i) Show that...
- 10N.1.sl.TZ0.8b(i) and (ii): The line (AC) has equation \({\boldsymbol{r}} = {\boldsymbol{u}} + s{\boldsymbol{v}}\) . (i) ...
- 10N.1.sl.TZ0.8c: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Show that...
- 10N.1.sl.TZ0.8d: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Hence find the...
- 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
- 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
- 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
- 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
- 09M.2.sl.TZ1.5: Two lines with equations...
- 09M.1.sl.TZ2.10c: Let B be the point of intersection of lines \({L_1}\) and \({L_2}\) . (i) Show that...
- 11N.1.sl.TZ0.8a(i) and (ii): (i) Find \(\overrightarrow {{\rm{PQ}}} \) . (ii) Hence write down a vector equation for...
- 11N.1.sl.TZ0.8b(i) and (ii): (i) Find the value of p . (ii) Given that \({L_2}\) passes through...
- 11N.1.sl.TZ0.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point A. Find the x-coordinate of A.
- 11M.2.sl.TZ2.8a: Find \(\overrightarrow {{\rm{AB}}} \) .
- 11M.2.sl.TZ2.8b: Find an equation for \({L_1}\) in the form...
- 11M.2.sl.TZ2.8c: Find the angle between \({L_1}\) and \({L_2}\) .
- 11M.2.sl.TZ2.8d: The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.
- 13M.1.sl.TZ1.8c: The line \({L_1}\) intersects the line \({L_2}\) at point Q. Find the \(x\)-coordinate of Q.
- 13M.2.sl.TZ2.4: Line \({L_1}\) has...
- 14M.1.sl.TZ1.8c: Find the coordinates of P.
- 14M.1.sl.TZ2.9d: The two airplanes collide at the point \((-23, 20, 28)\). How long after Ryan’s airplane takes...
- 13N.2.sl.TZ0.9a: Find the coordinates of \({\text{P}}\).
- 15N.1.sl.TZ0.9b: A line \({L_2}\) has equation...
- 17M.1.sl.TZ1.8a.ii: Hence, write down a vector equation for \({L_1}\).
- 17M.1.sl.TZ1.8b: A second line \({L_2}\), has equation r =...
- 17M.1.sl.TZ1.8c: The lines \({L_1}\) and \({L_1}\) intersect at \(C(9,{\text{ }}13,{\text{ }}z)\). Find \(z\).
- 17M.1.sl.TZ1.8d.i: Find a unit vector in the direction of \({L_2}\).
- 17M.1.sl.TZ1.8d.ii: Hence or otherwise, find one point on \({L_2}\) which is \(\sqrt 5 \) units from C.
