DP Mathematics HL Questionbank
Topic 4 - Core: Vectors
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Description
The aim of this topic is to introduce the use of vectors in two and three dimensions, and to facilitate solving problems involving points, lines and planes.
Directly related questions
- 12M.2.hl.TZ2.11c: The plane \( \div \) is parallel to both the line in part (b) and the line...
- 12N.2.hl.TZ0.13d: Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and...
- 12N.2.hl.TZ0.13c: The line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and...
- 08M.1.hl.TZ2.11: Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0). (a) Calculate the cosine of...
- SPNone.2.hl.TZ0.10a: (i) Find \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \) . (ii) ...
- SPNone.2.hl.TZ0.13e: Consider the points \({\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)\) ,...
- 13M.2.hl.TZ1.11b: Show that \({L_1}\) and \({L_2}\) intersect at the point R(1, 2, 4).
- 13M.2.hl.TZ1.11e: Let S be a point on \({L_2}\) such that...
- 10N.1.hl.TZ0.7: Consider the plane with equation \(4x - 2y - z = 1\) and the line given by the parametric...
- 13M.1.hl.TZ2.11d: Find a vector equation of (AB).
- 11N.2.hl.TZ0.13c: The point P has coordinates (−2, 4, 1) , the point Q lies on \({\Pi _3}\) and PQ is perpendicular...
- 11M.1.hl.TZ1.4a: Write down expressions for \(\overrightarrow {{\text{AB}}} \) and...
- 09N.2.hl.TZ0.2: The vector equation of line \(l\) is given as...
- 09M.2.hl.TZ1.7: (a) If \(a = 4\) find the coordinates of the point of intersection of the three...
- 14M.1.hl.TZ1.12c: Show that an equation of the plane \({\mathit{\Pi }}\), containing the square OABC, is...
- 14N.1.hl.TZ0.12c: Prove that \(T\) lies on [\(BM\)].
- 17M.2.hl.TZ1.7: Find the Cartesian equation of plane Π containing the points...
- 16M.1.hl.TZ1.11b: (i) Show that \(L\) has direction...
- 18M.1.hl.TZ1.10e: Find the area of the triangle OPQ.
- 18M.1.hl.TZ2.9d: Find the vector equation of the straight line passing through M and normal to the plane \(\Pi...
- 12M.2.hl.TZ1.13f: A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around...
- 12N.1.hl.TZ0.9a: Find the coordinates of the common point of the paths of the two boats.
- 08N.1.hl.TZ0.10: Three distinct non-zero vectors are given by \(\overrightarrow {{\text{OA}}} \) = a,...
- 11M.2.hl.TZ2.11e: Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to...
- 10M.1.hl.TZ1.3: (a) Show that the two planes \[{\pi _1}:x + 2y - z = 1\] \[{\pi _2}:x + z = - 2\] are...
- 10M.1.hl.TZ1.6: Consider the vectors \(\overrightarrow {{\text{OA}}} \) = a, \(\overrightarrow {{\text{OB}}} \) =...
- 11M.1.hl.TZ1.11b: Find the Cartesian equation of the plane \(\prod \) that contains the face ABC.
- 09N.2.hl.TZ0.11: (a) Find the coordinates of the point \(A\) on \({l_1}\) and the point \(B\) on \({l_2}\)...
- 14M.1.hl.TZ1.12d: Find a vector equation of the line \(L\), through M, perpendicular to the plane \({\mathit{\Pi }}\).
- 13N.1.hl.TZ0.11e: Find the value of \(\alpha \) if all three planes contain \({L_1}\).
- 15M.2.hl.TZ2.11c: Find the distance between the two points on the curve where each tangent is parallel to the line...
- 14N.1.hl.TZ0.3: A point \(P\), relative to an origin \(O\), has position vector...
- 15N.1.hl.TZ0.13c: Show that the line segment [\(CP\)] also includes the point \(G\).
- 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
- 16M.1.hl.TZ1.11e: Find the coordinates of the two possible positions of \(P\).
- 16M.2.hl.TZ1.1a: Find \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \).
- 18M.1.hl.TZ2.9f.i: Find the coordinates of X, Y and Z.
- 12M.2.hl.TZ1.13c(i)(ii): (i) Find the Cartesian equation of \({\Pi _1}\) , the plane perpendicular to (AB)...
- 12M.2.hl.TZ2.11e: Find the angle between the line \(\frac{{x - 2}}{3} = \frac{{y + 5}}{4} = \frac{z}{2}\) and the...
- 08M.2.hl.TZ1.5: Find the vector equation of the line of intersection of the three planes represented by the...
- 08M.1.hl.TZ2.10: Given any two non-zero vectors a and b , show that \(|\)a \( \times \) b\({|^2}\) =...
- 11M.2.hl.TZ2.11g: A second plane, \({\prod _2}\) , has equation x − 2y + z = 3. Calculate the angle between the two...
- SPNone.2.hl.TZ0.8a: Show that OC = 2AB .
- 10N.2.hl.TZ0.12: The diagram shows a cube OABCDEFG. Let O be the origin, (OA) the x-axis, (OC) the y-axis...
- 10N.2.hl.TZ0.9: Consider the vectors a \( = \sin (2\alpha )\)i \( - \cos (2\alpha )\)j + k and b...
- 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
- 14M.1.hl.TZ2.12b: The line \({L_2}\) has Cartesian equation...
- 17M.2.hl.TZ2.7: Given that a \( \times \) b \( = \) b \( \times \) c \( \ne \) 0 prove that a \( + \) c \( = \)...
- 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
- 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
- 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
- 16M.2.hl.TZ1.1b: Hence find the area of the triangle OAB.
- 16M.1.hl.TZ2.10c: (i) Show that \(p = - 2\). (ii) If \(L\) intersects \(\Pi \) at \(z = - 1\), find the...
- 16M.2.hl.TZ2.9a: Show that (i) \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
- 18M.1.hl.TZ1.10b: Find the angle between the faces ABD and BCD.
- 18M.2.hl.TZ1.11c.i: Find an expression for the distance between the two submarines in terms of t.
- 18M.2.hl.TZ1.11c.iii: Find the distance between the two submarines at this time.
- 17M.2.hl.TZ2.9g: Find the acute angle between the planes Π\(_2\) and Π\(_3\).
- 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
- 12M.2.hl.TZ2.11b: Given that the system of equations can be solved, find the solutions in the form of a vector...
- 12N.2.hl.TZ0.13e: Let Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ...
- 08M.1.hl.TZ1.11: The points A, B, C have position vectors i + j + 2k , i + 2j + 3k , 3i + k respectively and lie...
- 08N.2.hl.TZ0.10: (a) Write the vector equations of the following lines in parametric...
- 11M.2.hl.TZ2.11d: Find the Cartesian equation of the plane, \({\prod _1}\) , containing the parallelogram PQRS.
- 11M.2.hl.TZ2.11f: Hence find the point on the plane that is closest to the origin.
- SPNone.2.hl.TZ0.8b: Find the position vectors of C, D and E in terms of a and b .
- 10M.1.hl.TZ2.3: The three vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\) are given...
- 10M.1.hl.TZ2.12b: Given two vectors p and q, (i) show that p\( \cdot \)p = \(|\)p\({|^2}\); (ii) hence,...
- 13M.1.hl.TZ2.11e: The point D on (AB) is such that \(\overrightarrow {{\text{OD}}} \) is perpendicular to...
- 11M.1.hl.TZ1.11a: Find the vectors \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{AC}}} \).
- 14M.1.hl.TZ1.12f: Find the coordinates of E, the reflection of the point D in the plane \({\mathit{\Pi }}\).
- 13N.2.hl.TZ0.9: A line \({L_1}\) has equation r =...
- 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
- 14M.1.hl.TZ1.12e: Find the coordinates of D, the point of intersection of the line \(L\) with the plane whose...
- 14M.1.hl.TZ2.12c: Find the Cartesian equation of the plane \({\Pi _1}\).
- 13N.1.hl.TZ0.11a: Find the vector \(\overrightarrow {{\text{CA}}} \times \overrightarrow {{\text{CB}}} \).
- 14N.1.hl.TZ0.12b: (i) Express \(\overrightarrow {{\text{RA}}} \) in terms of \(a\) and \(b\). (ii) Show...
- 14N.2.hl.TZ0.5a: Find the Cartesian equation of \(\pi \).
- 15N.2.hl.TZ0.10a: find the vectors \({{a}}\) and \({{b}}\).
- 15N.2.hl.TZ0.10b: Roderick is at a point \({\text{C}}(11,{\text{ }}9)\). During Ed’s walk from \(P\) to \(Q\)...
- 17M.2.hl.TZ2.9b: Determine whether or not the lines (OA) and (BC) intersect.
- 17M.2.hl.TZ2.9c: Find the Cartesian equation of the plane Π\(_1\), which passes through C and is perpendicular to...
- 17N.1.hl.TZ0.9c: Show that \(\mu = \frac{1}{{13}}\), and find the value of \(\lambda \).
- 16N.1.hl.TZ0.4a: Find a \( \times \) b.
- 16N.1.hl.TZ0.8b: determine the coordinates of the point of intersection P.
- 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}} = \) a,...
- 16M.1.hl.TZ2.1: The following system of equations represents three planes in space. \[x + 3y + z = -...
- 18M.2.hl.TZ1.11b.i: Show that submarine B travels in the same direction as originally planned.
- 12M.2.hl.TZ1.13e: A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around...
- 12N.2.hl.TZ0.13b: The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector...
- 11M.2.hl.TZ2.11c: Hence calculate the area of parallelogram PQRS.
- 10N.1.hl.TZ0.10: Let \(\alpha \) be the angle between the unit vectors a and b, where...
- 13M.1.hl.TZ2.11b: (i) Show that \(\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{CA}}} = \) −7i...
- 13M.1.hl.TZ2.11c: Find the Cartesian equation of the plane containing the triangle ABC.
- 09M.2.hl.TZ1.5: Find the angle between the lines \(\frac{{x - 1}}{2} = 1 - y = 2z\) and \(x = y = 3z\) .
- 09M.2.hl.TZ1.11: (a) Find the coordinates of P when \(t = 0\) . (b) Show that P moves along the line...
- 13N.1.hl.TZ0.11d: A second plane \({\Pi _2}\) is defined by the Cartesian equation \({\Pi _2}:4x - y - z = 4\)....
- 14N.1.hl.TZ0.12a: (i) Express \(\overrightarrow {{\text{AM}}} \) in terms of \(a\) and \(c\). (ii) Hence...
- 14N.2.hl.TZ0.1: Consider the two planes \({\pi _1}:4x + 2y - z = 8\) \({\pi _2}:x + 3y + 3z =...
- 17M.2.hl.TZ2.9d: Show that the line (BC) lies in the plane Π\(_1\).
- 16N.1.hl.TZ0.1: Find the coordinates of the point of intersection of the planes defined by the equations...
- 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
- 18M.1.hl.TZ2.9e: Find the Cartesian equation of \(\Pi \).
- 12M.2.hl.TZ1.13b: Find the Cartesian equation of \(\Pi \), the plane passing through A, B, and C.
- 12M.2.hl.TZ1.13d: Find the vector equation of L , the line of intersection of \({\Pi _1}\) and \({\Pi _2}\) , and...
- 12M.2.hl.TZ2.11d: The z-axis meets the plane \( \div \) at the point P. Find the coordinates of P.
- 12N.1.hl.TZ0.9b: Show that the boats do not collide.
- 12N.2.hl.TZ0.13a: Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .
- 08N.2.hl.TZ0.4: The angle between the vector a = i − 2j + 3k and the vector b = 3i − 2j + mk is 30° . Find the...
- 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
- 11M.2.hl.TZ2.11b: The vector product...
- 09M.1.hl.TZ2.10: (a) Show that a Cartesian equation of the line, \({l_1}\), containing points A(1, −1, 2) and...
- 13M.1.hl.TZ1.2b: Hence find the area of the triangle ABC.
- 13M.1.hl.TZ1.2a: Find \(\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} \).
- 13M.2.hl.TZ1.11a: Find a vector equation for the line, \({L_1}\), which passes through the points P and Q. The...
- 13M.2.hl.TZ1.11c: Find the acute angle between \({L_1}\) and \({L_2}\).
- 10M.2.hl.TZ1.11: A plane \(\pi \) has vector equation r = (−2i + 3j − 2k) + \(\lambda \)(2i + 3j + 2k) + \(\mu...
- 13M.1.hl.TZ2.11a: (i) Find the lengths of the sides of the triangle. (ii) Find \(\cos {\rm{B\hat AC}}\).
- 11N.2.hl.TZ0.13b: Show that the plane \({\Pi _3}\) which is perpendicular to \({\Pi _1}\) and contains L, has...
- 11M.2.hl.TZ1.10: Port A is defined to be the origin of a set of coordinate axes and port B is located at the point...
- 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}} = \) \(\boldsymbol{a}\) and...
- 15M.2.hl.TZ2.13b: Find the acute angle between the lines \({L_1}\) and \({L_2}\).
- 14N.2.hl.TZ0.5b: The line \({l_3}\) passing through the point \((4,{\text{ }}0,{\text{ }}8)\) is perpendicular to...
- 15N.1.hl.TZ0.13a: Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).
- 17M.1.hl.TZ1.5b: By using a suitable scalar product of two vectors, determine whether \({\rm{A\hat BC}}\) is acute...
- 17M.2.hl.TZ2.9e: Verify that 2j + k is perpendicular to the plane Π\(_2\).
- 16N.1.hl.TZ0.8a: find the value of \(a\);
- 16N.1.hl.TZ0.4b: Hence find the Cartesian equation of the plane containing the vectors a and b, and passing...
- 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).
- 16M.1.hl.TZ1.11d: Show that \({\text{AB}} = 3\sqrt 2 \).
- 16M.2.hl.TZ2.9b: Given that...
- 18M.1.hl.TZ1.10c: Find the Cartesian equation of \({\Pi _3}\).
- 18M.2.hl.TZ1.11b.ii: Find the value of t when submarine B passes through P.
- 12M.2.hl.TZ1.4: The planes \(2x + 3y - z = 5\) and \(x - y + 2z = k\) intersect in the line...
- 08M.2.hl.TZ1.3: A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector...
- 13M.2.hl.TZ1.11d: Let S be a point on \({L_2}\) such that...
- 11N.1.hl.TZ0.12b: The points A, B and C have position vectors \({\boldsymbol{a}}\), \({\boldsymbol{b}}\) and...
- 11N.1.hl.TZ0.12a: For non-zero vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\), show that (i) if...
- 11N.2.hl.TZ0.13a: Find the vector equation of L, the line of intersection of \({\Pi _1}\) and \({\Pi _2}\).
- 11M.1.hl.TZ1.4b: Hence prove that angle \({\text{A}}\hat {\rm{B}}{\text{C}}\) is a right angle.
- 09M.2.hl.TZ2.2: Given that...
- 14M.1.hl.TZ1.12g: (i) Find the angle \({\rm{O\hat DA}}\). (ii) State what this tells you about the solid...
- 13N.1.hl.TZ0.11b: Find an exact value for the area of the triangle ABC.
- 13N.1.hl.TZ0.11c: Show that the Cartesian equation of the plane \({\Pi _1}\), containing the triangle ABC, is...
- 13N.1.hl.TZ0.11f: Find conditions on \(\alpha \) and \(\beta \) if the plane \({\Pi _3}\) does not intersect with...
- 13N.2.hl.TZ0.7a: show that \(3{\sin ^2}\theta - 7\sin \theta + 2 = 0\);
- 15N.1.hl.TZ0.13b: (i) Find a vector equation of the line that passes through \(B\) and \(R\) in terms of...
- 15N.1.hl.TZ0.13d: The coordinates of the points \(A\), \(B\) and \(C\) are \((1,{\text{ }}3,{\text{ }}1)\),...
- 17M.1.hl.TZ1.5a: Find the area of the parallelogram ABCD.
- 17M.2.hl.TZ2.9a: Find the vector equation of the line (BC).
- 17M.2.hl.TZ2.9f: Find a vector perpendicular to the plane Π\(_3\).
- 17N.1.hl.TZ0.2a: Find a vector equation of the line L passing through the points A and B.
- 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
- 17N.1.hl.TZ0.2b: Find the coordinates of the point of intersection of the line L with the plane Π.
- 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
- 16M.1.hl.TZ1.11a: Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where...
- 16M.1.hl.TZ1.11c: Given the vector \(\overrightarrow {{\text{AB}}} \) is perpendicular to \(L\) find the value of...
- 16M.1.hl.TZ2.10a: Show that \(L\) is not perpendicular to \(\Pi \).
- 16M.1.hl.TZ2.10b: Given that \(L\) lies in the plane \(\Pi \), find the value of \(p\) and the value of \(q\).
- 09M.1.hl.TZ1.8: A triangle has vertices A(1, −1, 1), B(1, 1, 0) and C(−1, 1, −1) . Show that the area of the...
- 18M.1.hl.TZ1.10a: Find the Cartesian equation of the plane \({\Pi _1}\), passing through the points A , B and D.
- 18M.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.
- 18M.2.hl.TZ1.11c.ii: Find the value of t when the two submarines are closest together.
- 18M.1.hl.TZ2.1: The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ. Find cos θ.
- 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
- 12M.2.hl.TZ1.13a: Show that AB = AC and that \({\rm{B\hat AC}} = 60^\circ \).
- 11M.1.hl.TZ2.6b: Hence prove that \({\rm{A\hat CB}}\) is a right angle.
- 11M.2.hl.TZ2.11a: Find the coordinates of S.
- SPNone.1.hl.TZ0.8: The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a \( \times \) b = b...
- SPNone.2.hl.TZ0.10b: (i) Find the vector equation of the line \({L_1}\) containing the points A and B. (ii) ...
- 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i) Find the cosine of the angle...
- 14M.1.hl.TZ2.12a: Given the points A(1, 0, 4), B(2, 3, −1) and C(0, 1, − 2) , find the vector equation of the line...
- 14M.1.hl.TZ2.12d: (i) Find the value of \(k\). (ii) Find the point of intersection P of the line \({L_3}\)...
- 15M.2.hl.TZ2.13a: Show that the lines \({L_1}\) and \({L_2}\) are skew.
- 18M.1.hl.TZ1.10d: Show that P is the midpoint of AD.
- 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
- 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.
- 18M.1.hl.TZ2.9f.ii: Find YZ.
Sub sections and their related questions
4.1
- 11M.1.hl.TZ2.6a: Find an expression for \(\overrightarrow {{\text{CB}}} \) and for...
- 11M.2.hl.TZ2.11a: Find the coordinates of S.
- SPNone.2.hl.TZ0.8a: Show that OC = 2AB .
- SPNone.2.hl.TZ0.8b: Find the position vectors of C, D and E in terms of a and b .
- 10M.1.hl.TZ2.3: The three vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\) are given...
- 10N.1.hl.TZ0.10: Let \(\alpha \) be the angle between the unit vectors a and b, where...
- 13M.1.hl.TZ2.11a: (i) Find the lengths of the sides of the triangle. (ii) Find \(\cos {\rm{B\hat AC}}\).
- 11M.1.hl.TZ1.4a: Write down expressions for \(\overrightarrow {{\text{AB}}} \) and...
- 11M.1.hl.TZ1.11a: Find the vectors \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{AC}}} \).
- 14M.1.hl.TZ1.12b: Find the coordinates of M, the mid-point of [OB].
- 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}} = \) \(\boldsymbol{a}\) and...
- 14M.1.hl.TZ1.12a: Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\),...
- 14N.1.hl.TZ0.3: A point \(P\), relative to an origin \(O\), has position vector...
- 14N.1.hl.TZ0.12a: (i) Express \(\overrightarrow {{\text{AM}}} \) in terms of \(a\) and \(c\). (ii) Hence...
- 14N.1.hl.TZ0.12b: (i) Express \(\overrightarrow {{\text{RA}}} \) in terms of \(a\) and \(b\). (ii) Show...
- 14N.1.hl.TZ0.12c: Prove that \(T\) lies on [\(BM\)].
- 15M.2.hl.TZ2.11c: Find the distance between the two points on the curve where each tangent is parallel to the line...
- 15N.1.hl.TZ0.13a: Find \(\overrightarrow {{\text{BR}}} \) in terms of \({{a}}\), \({{b}}\) and \({{c}}\).
- 15N.2.hl.TZ0.10a: find the vectors \({{a}}\) and \({{b}}\).
- 16M.1.hl.TZ1.8: O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}} = \) a,...
- 16M.2.hl.TZ2.9a: Show that (i) \({\left| {\overrightarrow {{\text{OC}}} } \right|^2} = |\)a\({|^2} +...
- 16M.2.hl.TZ2.9b: Given that...
- 17N.1.hl.TZ0.9a.i: Find, in terms of a and b \(\overrightarrow {{\text{OF}}} \).
- 17N.1.hl.TZ0.9a.ii: Find, in terms of a and b \(\overrightarrow {{\text{AF}}} \).
- 17N.1.hl.TZ0.9b.i: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\lambda \);
- 17N.1.hl.TZ0.9b.ii: Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of a, b and \(\mu \).
- 17N.1.hl.TZ0.9c: Show that \(\mu = \frac{1}{{13}}\), and find the value of \(\lambda \).
- 17N.1.hl.TZ0.9d: Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of a and b only.
- 17N.1.hl.TZ0.9e: Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of...
- 18M.2.hl.TZ1.11a: Show that the two submarines would collide at a point P and write down the coordinates of P.
- 18M.2.hl.TZ1.11b.i: Show that submarine B travels in the same direction as originally planned.
- 18M.2.hl.TZ1.11b.ii: Find the value of t when submarine B passes through P.
- 18M.1.hl.TZ2.9a.i: Explain why ABCD is a parallelogram.
- 18M.1.hl.TZ2.9a.ii: Using vector algebra, show that...
- 18M.1.hl.TZ2.9b: Show that p = 1, q = 1 and r = 4.
- 18M.1.hl.TZ2.9c: Find the area of the parallelogram ABCD.
4.2
- 12M.2.hl.TZ1.13a: Show that AB = AC and that \({\rm{B\hat AC}} = 60^\circ \).
- 12M.2.hl.TZ1.13f: A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around...
- 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
- 12N.2.hl.TZ0.13e: Let Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ...
- 08M.1.hl.TZ2.10: Given any two non-zero vectors a and b , show that \(|\)a \( \times \) b\({|^2}\) =...
- 08N.1.hl.TZ0.10: Three distinct non-zero vectors are given by \(\overrightarrow {{\text{OA}}} \) = a,...
- 08N.2.hl.TZ0.4: The angle between the vector a = i − 2j + 3k and the vector b = 3i − 2j + mk is 30° . Find the...
- 11M.1.hl.TZ2.6b: Hence prove that \({\rm{A\hat CB}}\) is a right angle.
- 13M.2.hl.TZ1.11c: Find the acute angle between \({L_1}\) and \({L_2}\).
- 13M.2.hl.TZ1.11e: Let S be a point on \({L_2}\) such that...
- 10M.1.hl.TZ1.6: Consider the vectors \(\overrightarrow {{\text{OA}}} \) = a, \(\overrightarrow {{\text{OB}}} \) =...
- 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i) Find the cosine of the angle...
- 10M.1.hl.TZ2.12b: Given two vectors p and q, (i) show that p\( \cdot \)p = \(|\)p\({|^2}\); (ii) hence,...
- 10N.2.hl.TZ0.9: Consider the vectors a \( = \sin (2\alpha )\)i \( - \cos (2\alpha )\)j + k and b...
- 13M.1.hl.TZ2.11a: (i) Find the lengths of the sides of the triangle. (ii) Find \(\cos {\rm{B\hat AC}}\).
- 11N.1.hl.TZ0.12a: For non-zero vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\), show that (i) if...
- 11N.1.hl.TZ0.12b: The points A, B and C have position vectors \({\boldsymbol{a}}\), \({\boldsymbol{b}}\) and...
- 11M.1.hl.TZ1.4b: Hence prove that angle \({\text{A}}\hat {\rm{B}}{\text{C}}\) is a right angle.
- 09M.2.hl.TZ2.2: Given that...
- 14M.1.hl.TZ1.12g: (i) Find the angle \({\rm{O\hat DA}}\). (ii) State what this tells you about the solid...
- 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}} = \) \(\boldsymbol{a}\) and...
- 13N.2.hl.TZ0.7a: show that \(3{\sin ^2}\theta - 7\sin \theta + 2 = 0\);
- 15N.2.hl.TZ0.10b: Roderick is at a point \({\text{C}}(11,{\text{ }}9)\). During Ed’s walk from \(P\) to \(Q\)...
- 16M.1.hl.TZ1.11a: Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where...
- 17M.1.hl.TZ1.5b: By using a suitable scalar product of two vectors, determine whether \({\rm{A\hat BC}}\) is acute...
- 17M.2.hl.TZ2.9g: Find the acute angle between the planes Π\(_2\) and Π\(_3\).
- 18M.1.hl.TZ2.1: The acute angle between the vectors 3i − 4j − 5k and 5i − 4j + 3k is denoted by θ. Find cos θ.
4.3
- 12M.2.hl.TZ1.4: The planes \(2x + 3y - z = 5\) and \(x - y + 2z = k\) intersect in the line...
- 12M.2.hl.TZ1.13d: Find the vector equation of L , the line of intersection of \({\Pi _1}\) and \({\Pi _2}\) , and...
- 12M.2.hl.TZ1.13e: A methane molecule consists of a carbon atom with four hydrogen atoms symmetrically placed around...
- 12M.2.hl.TZ2.11b: Given that the system of equations can be solved, find the solutions in the form of a vector...
- 12N.2.hl.TZ0.13b: The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector...
- 11M.2.hl.TZ2.11e: Write down the vector equation of the line through the origin (0, 0, 0) that is perpendicular to...
- 11M.2.hl.TZ2.11f: Hence find the point on the plane that is closest to the origin.
- SPNone.2.hl.TZ0.10b: (i) Find the vector equation of the line \({L_1}\) containing the points A and B. (ii) ...
- 13M.2.hl.TZ1.11a: Find a vector equation for the line, \({L_1}\), which passes through the points P and Q. The...
- 13M.2.hl.TZ1.11c: Find the acute angle between \({L_1}\) and \({L_2}\).
- 13M.2.hl.TZ1.11d: Let S be a point on \({L_2}\) such that...
- 13M.2.hl.TZ1.11e: Let S be a point on \({L_2}\) such that...
- 13M.1.hl.TZ2.11d: Find a vector equation of (AB).
- 13M.1.hl.TZ2.11e: The point D on (AB) is such that \(\overrightarrow {{\text{OD}}} \) is perpendicular to...
- 11N.2.hl.TZ0.13a: Find the vector equation of L, the line of intersection of \({\Pi _1}\) and \({\Pi _2}\).
- 11M.2.hl.TZ1.10: Port A is defined to be the origin of a set of coordinate axes and port B is located at the point...
- 09N.2.hl.TZ0.2: The vector equation of line \(l\) is given as...
- 09N.2.hl.TZ0.11: (a) Find the coordinates of the point \(A\) on \({l_1}\) and the point \(B\) on \({l_2}\)...
- 09M.2.hl.TZ1.5: Find the angle between the lines \(\frac{{x - 1}}{2} = 1 - y = 2z\) and \(x = y = 3z\) .
- 09M.2.hl.TZ1.11: (a) Find the coordinates of P when \(t = 0\) . (b) Show that P moves along the line...
- 14M.1.hl.TZ1.12d: Find a vector equation of the line \(L\), through M, perpendicular to the plane \({\mathit{\Pi }}\).
- 14M.1.hl.TZ2.12a: Given the points A(1, 0, 4), B(2, 3, −1) and C(0, 1, − 2) , find the vector equation of the line...
- 13N.1.hl.TZ0.11d: A second plane \({\Pi _2}\) is defined by the Cartesian equation \({\Pi _2}:4x - y - z = 4\)....
- 13N.2.hl.TZ0.9: A line \({L_1}\) has equation r =...
- 14N.2.hl.TZ0.5a: Find the Cartesian equation of \(\pi \).
- 15M.2.hl.TZ2.13a: Show that the lines \({L_1}\) and \({L_2}\) are skew.
- 15M.2.hl.TZ2.13b: Find the acute angle between the lines \({L_1}\) and \({L_2}\).
- 15N.1.hl.TZ0.13b: (i) Find a vector equation of the line that passes through \(B\) and \(R\) in terms of...
- 15N.1.hl.TZ0.13c: Show that the line segment [\(CP\)] also includes the point \(G\).
- 15N.2.hl.TZ0.10a: find the vectors \({{a}}\) and \({{b}}\).
- 16M.1.hl.TZ2.10a: Show that \(L\) is not perpendicular to \(\Pi \).
- 16M.1.hl.TZ2.10b: Given that \(L\) lies in the plane \(\Pi \), find the value of \(p\) and the value of \(q\).
- 16M.1.hl.TZ1.11b: (i) Show that \(L\) has direction...
- 16N.1.hl.TZ0.8a: find the value of \(a\);
- 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).
- 17M.2.hl.TZ2.9a: Find the vector equation of the line (BC).
- 17N.1.hl.TZ0.2a: Find a vector equation of the line L passing through the points A and B.
- 18M.1.hl.TZ2.9d: Find the vector equation of the straight line passing through M and normal to the plane \(\Pi...
4.4
- 12N.1.hl.TZ0.9a: Find the coordinates of the common point of the paths of the two boats.
- 12N.1.hl.TZ0.9b: Show that the boats do not collide.
- 12N.2.hl.TZ0.13c: The line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and...
- 08M.1.hl.TZ2.11: Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0). (a) Calculate the cosine of...
- SPNone.2.hl.TZ0.10b: (i) Find the vector equation of the line \({L_1}\) containing the points A and B. (ii) ...
- 13M.2.hl.TZ1.11b: Show that \({L_1}\) and \({L_2}\) intersect at the point R(1, 2, 4).
- 14M.1.hl.TZ2.12b: The line \({L_2}\) has Cartesian equation...
- 15M.2.hl.TZ2.13a: Show that the lines \({L_1}\) and \({L_2}\) are skew.
- 15N.1.hl.TZ0.13b: (i) Find a vector equation of the line that passes through \(B\) and \(R\) in terms of...
- 15N.1.hl.TZ0.13c: Show that the line segment [\(CP\)] also includes the point \(G\).
- 15N.2.hl.TZ0.10b: Roderick is at a point \({\text{C}}(11,{\text{ }}9)\). During Ed’s walk from \(P\) to \(Q\)...
- 17M.2.hl.TZ2.9b: Determine whether or not the lines (OA) and (BC) intersect.
- 18M.2.hl.TZ1.11c.i: Find an expression for the distance between the two submarines in terms of t.
- 18M.2.hl.TZ1.11c.ii: Find the value of t when the two submarines are closest together.
- 18M.2.hl.TZ1.11c.iii: Find the distance between the two submarines at this time.
4.5
- 12N.2.hl.TZ0.13d: Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and...
- 08M.1.hl.TZ2.10: Given any two non-zero vectors a and b , show that \(|\)a \( \times \) b\({|^2}\) =...
- 11M.2.hl.TZ2.11b: The vector product...
- 11M.2.hl.TZ2.11c: Hence calculate the area of parallelogram PQRS.
- 09M.1.hl.TZ1.8: A triangle has vertices A(1, −1, 1), B(1, 1, 0) and C(−1, 1, −1) . Show that the area of the...
- SPNone.1.hl.TZ0.8: The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a \( \times \) b = b...
- SPNone.2.hl.TZ0.10a: (i) Find \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \) . (ii) ...
- SPNone.2.hl.TZ0.13e: Consider the points \({\text{A}}\left( {a{\text{ }},{\text{ }}f(a)} \right)\) ,...
- 13M.1.hl.TZ1.2a: Find \(\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} \).
- 13M.1.hl.TZ1.2b: Hence find the area of the triangle ABC.
- 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i) Find the cosine of the angle...
- 10N.2.hl.TZ0.9: Consider the vectors a \( = \sin (2\alpha )\)i \( - \cos (2\alpha )\)j + k and b...
- 10N.2.hl.TZ0.12: The diagram shows a cube OABCDEFG. Let O be the origin, (OA) the x-axis, (OC) the y-axis...
- 13M.1.hl.TZ2.11b: (i) Show that \(\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{CA}}} = \) −7i...
- 11N.1.hl.TZ0.12a: For non-zero vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\), show that (i) if...
- 11N.1.hl.TZ0.12b: The points A, B and C have position vectors \({\boldsymbol{a}}\), \({\boldsymbol{b}}\) and...
- 11M.1.hl.TZ1.11b: Find the Cartesian equation of the plane \(\prod \) that contains the face ABC.
- 13N.1.hl.TZ0.11b: Find an exact value for the area of the triangle ABC.
- 13N.1.hl.TZ0.11a: Find the vector \(\overrightarrow {{\text{CA}}} \times \overrightarrow {{\text{CB}}} \).
- 15N.1.hl.TZ0.13d: The coordinates of the points \(A\), \(B\) and \(C\) are \((1,{\text{ }}3,{\text{ }}1)\),...
- 16M.2.hl.TZ1.1a: Find \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \).
- 16M.2.hl.TZ1.1b: Hence find the area of the triangle OAB.
- 16N.1.hl.TZ0.4a: Find a \( \times \) b.
- 17M.1.hl.TZ1.5a: Find the area of the parallelogram ABCD.
- 17M.2.hl.TZ2.7: Given that a \( \times \) b \( = \) b \( \times \) c \( \ne \) 0 prove that a \( + \) c \( = \)...
4.6
- 12M.2.hl.TZ1.4: The planes \(2x + 3y - z = 5\) and \(x - y + 2z = k\) intersect in the line...
- 12M.2.hl.TZ1.13b: Find the Cartesian equation of \(\Pi \), the plane passing through A, B, and C.
- 12M.2.hl.TZ1.13c(i)(ii): (i) Find the Cartesian equation of \({\Pi _1}\) , the plane perpendicular to (AB)...
- 12M.2.hl.TZ2.11c: The plane \( \div \) is parallel to both the line in part (b) and the line...
- 12N.2.hl.TZ0.13d: Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and...
- 08M.1.hl.TZ2.11: Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0). (a) Calculate the cosine of...
- 11M.2.hl.TZ2.11d: Find the Cartesian equation of the plane, \({\prod _1}\) , containing the parallelogram PQRS.
- SPNone.2.hl.TZ0.10a: (i) Find \(\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} \) . (ii) ...
- 10M.1.hl.TZ1.3: (a) Show that the two planes \[{\pi _1}:x + 2y - z = 1\] \[{\pi _2}:x + z = - 2\] are...
- 10M.2.hl.TZ1.11: A plane \(\pi \) has vector equation r = (−2i + 3j − 2k) + \(\lambda \)(2i + 3j + 2k) + \(\mu...
- 10M.1.hl.TZ2.12a: Consider the vectors a = 6i + 3j + 2k, b = −3j + 4k. (i) Find the cosine of the angle...
- 13M.1.hl.TZ2.11c: Find the Cartesian equation of the plane containing the triangle ABC.
- 11N.2.hl.TZ0.13b: Show that the plane \({\Pi _3}\) which is perpendicular to \({\Pi _1}\) and contains L, has...
- 11M.1.hl.TZ1.11b: Find the Cartesian equation of the plane \(\prod \) that contains the face ABC.
- 09N.2.hl.TZ0.2: The vector equation of line \(l\) is given as...
- 09N.2.hl.TZ0.11: (a) Find the coordinates of the point \(A\) on \({l_1}\) and the point \(B\) on \({l_2}\)...
- 14M.1.hl.TZ1.12c: Show that an equation of the plane \({\mathit{\Pi }}\), containing the square OABC, is...
- 14M.1.hl.TZ2.12d: (i) Find the value of \(k\). (ii) Find the point of intersection P of the line \({L_3}\)...
- 13N.1.hl.TZ0.11c: Show that the Cartesian equation of the plane \({\Pi _1}\), containing the triangle ABC, is...
- 14M.1.hl.TZ2.12c: Find the Cartesian equation of the plane \({\Pi _1}\).
- 14N.2.hl.TZ0.5a: Find the Cartesian equation of \(\pi \).
- 16N.1.hl.TZ0.4b: Hence find the Cartesian equation of the plane containing the vectors a and b, and passing...
- 17M.2.hl.TZ1.7: Find the Cartesian equation of plane Π containing the points...
- 17M.2.hl.TZ2.9c: Find the Cartesian equation of the plane Π\(_1\), which passes through C and is perpendicular to...
- 17M.2.hl.TZ2.9d: Show that the line (BC) lies in the plane Π\(_1\).
- 17M.2.hl.TZ2.9e: Verify that 2j + k is perpendicular to the plane Π\(_2\).
- 17M.2.hl.TZ2.9f: Find a vector perpendicular to the plane Π\(_3\).
- 18M.1.hl.TZ1.10a: Find the Cartesian equation of the plane \({\Pi _1}\), passing through the points A , B and D.
- 18M.1.hl.TZ1.10c: Find the Cartesian equation of \({\Pi _3}\).
- 18M.1.hl.TZ2.9e: Find the Cartesian equation of \(\Pi \).
4.7
- 12M.2.hl.TZ1.4: The planes \(2x + 3y - z = 5\) and \(x - y + 2z = k\) intersect in the line...
- 12M.2.hl.TZ2.11d: The z-axis meets the plane \( \div \) at the point P. Find the coordinates of P.
- 12M.2.hl.TZ2.11e: Find the angle between the line \(\frac{{x - 2}}{3} = \frac{{y + 5}}{4} = \frac{z}{2}\) and the...
- 12N.2.hl.TZ0.13a: Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .
- 08M.1.hl.TZ1.11: The points A, B, C have position vectors i + j + 2k , i + 2j + 3k , 3i + k respectively and lie...
- 08M.2.hl.TZ1.3: A ray of light coming from the point (−1, 3, 2) is travelling in the direction of vector...
- 08M.2.hl.TZ1.5: Find the vector equation of the line of intersection of the three planes represented by the...
- 08N.2.hl.TZ0.10: (a) Write the vector equations of the following lines in parametric...
- 11M.2.hl.TZ2.11g: A second plane, \({\prod _2}\) , has equation x − 2y + z = 3. Calculate the angle between the two...
- 10M.2.hl.TZ1.11: A plane \(\pi \) has vector equation r = (−2i + 3j − 2k) + \(\lambda \)(2i + 3j + 2k) + \(\mu...
- 10N.1.hl.TZ0.7: Consider the plane with equation \(4x - 2y - z = 1\) and the line given by the parametric...
- 10N.2.hl.TZ0.12: The diagram shows a cube OABCDEFG. Let O be the origin, (OA) the x-axis, (OC) the y-axis...
- 11N.2.hl.TZ0.13c: The point P has coordinates (−2, 4, 1) , the point Q lies on \({\Pi _3}\) and PQ is perpendicular...
- 09M.2.hl.TZ1.7: (a) If \(a = 4\) find the coordinates of the point of intersection of the three...
- 14M.1.hl.TZ1.12f: Find the coordinates of E, the reflection of the point D in the plane \({\mathit{\Pi }}\).
- 14M.1.hl.TZ2.12d: (i) Find the value of \(k\). (ii) Find the point of intersection P of the line \({L_3}\)...
- 13N.1.hl.TZ0.11f: Find conditions on \(\alpha \) and \(\beta \) if the plane \({\Pi _3}\) does not intersect with...
- 14M.1.hl.TZ1.12e: Find the coordinates of D, the point of intersection of the line \(L\) with the plane whose...
- 14M.1.hl.TZ2.12c: Find the Cartesian equation of the plane \({\Pi _1}\).
- 13N.1.hl.TZ0.11e: Find the value of \(\alpha \) if all three planes contain \({L_1}\).
- 14N.2.hl.TZ0.1: Consider the two planes \({\pi _1}:4x + 2y - z = 8\) \({\pi _2}:x + 3y + 3z =...
- 14N.2.hl.TZ0.5b: The line \({l_3}\) passing through the point \((4,{\text{ }}0,{\text{ }}8)\) is perpendicular to...
- 16M.1.hl.TZ2.1: The following system of equations represents three planes in space. \[x + 3y + z = -...
- 16M.1.hl.TZ2.10c: (i) Show that \(p = - 2\). (ii) If \(L\) intersects \(\Pi \) at \(z = - 1\), find the...
- 16M.1.hl.TZ1.11c: Given the vector \(\overrightarrow {{\text{AB}}} \) is perpendicular to \(L\) find the value of...
- 16M.1.hl.TZ1.11d: Show that \({\text{AB}} = 3\sqrt 2 \).
- 16M.1.hl.TZ1.11e: Find the coordinates of the two possible positions of \(P\).
- 16N.1.hl.TZ0.1: Find the coordinates of the point of intersection of the planes defined by the equations...
- 16N.1.hl.TZ0.8b: determine the coordinates of the point of intersection P.
- 17N.1.hl.TZ0.2b: Find the coordinates of the point of intersection of the line L with the plane Π.
- 18M.1.hl.TZ1.10b: Find the angle between the faces ABD and BCD.
- 18M.1.hl.TZ1.10d: Show that P is the midpoint of AD.
- 18M.1.hl.TZ1.10e: Find the area of the triangle OPQ.
- 18M.1.hl.TZ2.9f.i: Find the coordinates of X, Y and Z.
- 18M.1.hl.TZ2.9f.ii: Find YZ.
