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AHL 3.16—Vector product - IB Questionbank | RevisionDojo

Practice AHL 3.16—Vector product 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

The position vectors of points $P$ and $Q$ are $p=3 i+2 j+k$ and $q=i+3 j-2 k$

Find the vector product $p \times q$

[4]

Using your answer to part 1., or otherwise, find the area of the parallelogram with two sides $\overrightarrow{O P}$ and $\overrightarrow{O Q}$

[5]

Question 3

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 4

HLPaper 1

Three points $A(1,2,3)$ , $B(3,-1,4)$ , and $C(2,0,6)$ define a plane $\Pi$ .

Find the vector equation of the plane $\Pi$ .

[3]

Find the Cartesian equation of $\Pi$ .

[2]

Verify that point $D(3,1,1)$ does not lie on $\Pi$ , and find the perpendicular distance from $D$ to $\Pi$ .

[3]

Find the acute angle between the plane $\Pi$ and the line
$L:\ \mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+t\begin{pmatrix}1\\-3\\-2\end{pmatrix}.$

[4]

Find a unit vector lying in the plane $\Pi$ and perpendicular to the line of intersection of $\Pi$ with the $xz$ -plane.

[3]

Question 5

HLPaper 1

The coordinates of the points $P, Q, R$ and $S$ are $(4,1,-1),(3,3,5),(1,0,2 c)$ and $(1,1,2)$ respectively. The vectors $\overrightarrow{Q R}$ and $\overrightarrow{P R}$ are orthogonal (perpendicular).

Find the value of $c$ .

[9]

Calculate $\overrightarrow{P S} \times \overrightarrow{P R}$

[4]

Find the vector equation of the line I which passes through the point $Q$ and is parallel to the vector $P R$ .

[3]

Question 6

HLPaper 1

Consider two vectors $\vec{a} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \\ 2 \\ -1 \end{pmatrix}$

Calculate $\vec{a} \times \vec{b}$

[3]

Question 7

HLPaper 2

A construction company is analyzing the forces acting on a beam supported by two cables. The forces exerted by the cables are represented by vectors. The company needs to ensure that the angle between the two force vectors is within a safe range to prevent structural failure.

Given the force vectors $\mathbf{F_1} = \begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix}$ and $\mathbf{F_2} = \begin{pmatrix} 5 \\ 0 \\ 12 \end{pmatrix}$ , calculate the dot product $\mathbf{F_1} \cdot \mathbf{F_2}$ .

[2]

Using the dot product calculated, determine the cosine of the angle $\theta$ between the two vectors $\mathbf{F_1}$ and $\mathbf{F_2}$ .

[3]

Determine if the angle between the two vectors is within the safe range of $30^\circ$ to $60^\circ$ .

[3]

Question 8

HLPaper 1

Given that $a=i+2 j-k, b=-3 i+2 j+2 k$ and $c=2 i-3 j+4 k$ . Required to find $(a \times b) \cdot c$

Required to find $(a \times b) \cdot c$

$(a \times b) \cdot c=$

[8]

Question 9

HLPaper 2

Consider two planes $\Pi_1$ and $\Pi_2$ with equations: $\Pi_1: 2x + 3y - z = 4$ $\Pi_2: -x + 2y + 2z = 6$

Find the angle between the two planes.

[3]

Find a point that lies on both planes.

[3]

Find a direction vector of the line of intersection of these planes.

[4]

Question 10

HLPaper 1

Consider the line $L_1$ given by the equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$ and the plane $\pi$ given by the equation $2x - y + z = 5$ .

Find the point of intersection of the line $L_1$ and the plane $\pi$ .

[4]

Determine if the line $L_1$ is parallel, perpendicular, or neither to the normal of the plane $\pi$ .

[4]

AHL 3.16—Vector product Questionbank