DP Mathematics SL Questionbank

• Syllabus

Topic 4 - Vectors

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.