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 $Q$...