DP Mathematics HL Questionbank

• Syllabus

4.2

Sub sections and their related questions

The definition of the scalar product of two vectors.

• 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...
• 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...
• 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.TZ2.12b: Given two vectors p and q, (i)     show that p\( \cdot \)p = \(|\)p\({|^2}\); (ii)     hence,...
• 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.
• 14M.1.hl.TZ2.6: PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}}  = \) \(\boldsymbol{a}\) and...
• 13N.2.hl.TZ0.7a: show that \(3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0\);
• 16M.1.hl.TZ1.11a: Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where...
• 17M.2.hl.TZ2.9g: Find the acute angle between the planes Π\(_2\) and Π\(_3\).

Properties of the scalar product: \({\boldsymbol{v}} \cdot {\boldsymbol{w}} = {\boldsymbol{w}} \cdot {\boldsymbol{v}}\) ; \({\boldsymbol{u}} \cdot \left( {{\mathbf{v}} + {\boldsymbol{w}}} \right) = {\boldsymbol{u}} \cdot {\boldsymbol{v}} + {\boldsymbol{u}} \cdot {\boldsymbol{w}}\) ; \(\left( {k{\boldsymbol{v}}} \right) \cdot {\boldsymbol{w}} = k\left( {{\boldsymbol{v}} \cdot {\boldsymbol{w}}} \right)\) ; \({\boldsymbol{v}} \cdot {\boldsymbol{v}} = {\left| {\boldsymbol{v}} \right|^2}\) .

• 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...
• 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,...
• 10M.1.hl.TZ1.6: Consider the vectors \(\overrightarrow {{\text{OA}}} \) = a, \(\overrightarrow {{\text{OB}}} \) =...
• 13M.1.hl.TZ2.11a: (i)     Find the lengths of the sides of the triangle. (ii)     Find \(\cos {\rm{B\hat AC}}\).
• 16M.1.hl.TZ1.11a: Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where...
• 17M.2.hl.TZ2.9g: Find the acute angle between the planes Π\(_2\) and Π\(_3\).

The angle between two vectors.

• 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...
• 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...
• 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...
• 13M.1.hl.TZ2.11a: (i)     Find the lengths of the sides of the triangle. (ii)     Find \(\cos {\rm{B\hat AC}}\).
• 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...
• 16M.1.hl.TZ1.11a: Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where...
• 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.

Perpendicular vectors; parallel vectors.

• 12M.1.hl.TZ2.2: Find the values of x for which the vectors...
• 15N.2.hl.TZ0.10b: Roderick is at a point \({\text{C}}(11,{\text{ }}9)\). During Ed’s walk from \(P\) to \(Q\)...