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AHL 3.12—Vector definitions - IB Questionbank | RevisionDojo

Practice AHL 3.12—Vector definitions with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Given vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$ , and a vector $\mathbf{w}$ that is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ .

Find a vector $\mathbf{w}$ that is perpendicular to both $\mathbf{u}$ and $\mathbf{v}$ .

[5]

Question 2

HLPaper 1

Consider the plane $\Pi_1$ given by the equation $x + 2y - z = 4$ and the plane $\Pi_2$ given by the equation $3x - y + 4z = 10$ .

Find the line of intersection of the planes $\Pi_1$ and $\Pi_2$ .

[4]

Verify if the line of intersection found is perpendicular to the normal of either plane.

[3]

Question 3

HLPaper 1

In this question, all lengths are in metres and time is in seconds. Consider two particles, $A_1$ and $A_2$ , which start to move at the same time. Particle $A_1$ moves in a straight line such that its displacement from a fixed-point is given by $s(t) = 10 - \frac{7}{4}t^2$ , for $t \geq 0$ .

Find an expression for the velocity of ${A}_1$ at time ${t}$ .

[2]

Particle $A_2$ also moves in a straight line. The position of $A_2$ is given by $r=-\frac{1}{6}+t\left(\begin{array}{c}4\\-3\end{array}\right).$ The speed of $A_1$ is greater than the speed of $A_2$ when $t>q$ . Find the value of $q$ .

[5]

Question 4

HLPaper 1

In the following diagram OBCA is a square of side 5 cm while ACDE is a rectangle of length 10 cm and width 5 cm . Let $a=\overrightarrow{O A}$ and $b=\overrightarrow{O B}$

Express the following vectors in terms of $a$ and $b$

$\overrightarrow{O C}$

[1]

$\overrightarrow{O D}$

[1]

$\overrightarrow{A D}$

[1]

$\overrightarrow{B A}$

[1]

Question 5

HLPaper 1

Let $\quad a=\left(\begin{array}{l}2 \\ 1 \\ 2\end{array}\right)_{\text {and }} \quad b=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)_{\text {Find : }}$

Magnitude of $a$

[2]

Magnitude of $b$

[2]

Unit vector corresponding to $a$

[2]

Unit vector corresponding to $b$

[2]

Question 6

HLPaper 2

In three-dimensional space, points are defined as
$A(2,1,0), \qquad B(5,3,-2), \qquad C(4,7,3).$

Find, in component form, the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

[2]

Compute the scalar product $\overrightarrow{AB}\cdot\overrightarrow{AC}$ and determine the angle $\angle BAC$ to 3 s.f.

[4]

Find the unit vector in the direction of $\overrightarrow{AB}+\overrightarrow{AC}$ .

[3]

Find the perpendicular distance from $A$ to the line through $B$ and $C$ .

[4]

Let $P$ be the foot of the perpendicular from $A$ onto the line $BC$ .
Find the coordinates of $P$ .

[4]

Question 7

HLPaper 1

In the triangle OAB , let $a=\overrightarrow{O A}$ and $b=\overrightarrow{O B}$ .

It is also given that M is the midpoint of AB , while N divides OA in ratio $\mathrm{ON}: \mathrm{NA}=2: 1$ . Express the following vectors in terms of $a$ and $b$ .

$\overrightarrow{A B}$

[1]

$\overrightarrow{O M}$

[1]

$\overrightarrow{N B}$

[1]

$\overrightarrow{M N}$

[1]

Question 8

HLPaper 1

Let $\mathbf{a}=(2,-1,3),\qquad \mathbf{b}=(k,4,-2),\qquad \mathbf{c}=(1,2,1),$ where $k\in\mathbb{R}$ .

Find the value of $k$ for which $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

[2]

For this value of $k$ , find the exact value of the angle between $\mathbf{b}$ and $\mathbf{c}$ .

[3]

Show that the three vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ are not coplanar.

[3]

Find a unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (for $k=5$ ), giving exact surd form.

[4]

Prove that the angle between $\mathbf{a}\times\mathbf{b}$ and $\mathbf{c}$ satisfies
$\cos\phi=\frac{[\mathbf{a},\mathbf{b},\mathbf{c}]}{|\mathbf{a}\times\mathbf{b}|\,|\mathbf{c}|}.$

[3]

Question 9

HLPaper 1

The vectors $u, v$ are given by $u=3 i+5 j, v=i-2 j$ . Find scalars $a, b$ such that $a(u+v)=8 i+(b-2) j$

[4]

Question 10

HLPaper 1

In the following diagram OBCA is a square of side 5 cm while ACDE is a rectangle of length 10 cm and width 5 cm . Let $a=\overrightarrow{O A}$ and $b=\overrightarrow{O B}$

Express the following vectors in terms of $a$ and $b$

$\overrightarrow{A B}$

[1]

$\overrightarrow{C E}$

[1]

$\overrightarrow{B E}$

[1]

$\overrightarrow{E C}$

[1]

AHL 3.12—Vector definitions Questionbank