AHL 3.13—Scalar (dot) product Questionbank

Practice AHL 3.13—Scalar (dot) 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

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 2

HLPaper 1

Find the angle between the following vectors $a$ and $b$ , giving your answer to the nearest degree.

$a=-4 i-2 j, b=i-7 j$

[4]

Question 3

HLPaper 2

In 3D space, the points are
$A(3,1,-1),\quad B(7,4,0),\quad C(5,-2,6).$

Find the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ .

[3]

Compute $\overrightarrow{AB}\cdot\overrightarrow{AC}$ and hence find $\angle BAC$ 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 $BC$ .

[3]

Question 4

HLPaper 2

A triangle has its vertices at $A(-1,3,2), B(3,6,1)$ and $C(-4,4,3)$

Find the value of $\overrightarrow{A B} \cdot \overrightarrow{A C}$

[4]

Show that, to three significant figures $\cos \angle B A C=-0.591$

[4]

Question 5

HLPaper 1

The vectors $\binom{2 x}{x-3}$ and $\binom{x+1}{5}$ are perpendicular for two values of $x$ .

Find the two values of $x$ .

[4]

Question 6

HLPaper 1

Consider two vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$ .

Calculate the dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ .

[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

Consider a circle with a diameter $XY$ , where $X$ has coordinates $(1, 4, 0)$ and $Y$ has coordinates $(-3, 2, -4)$ . The circle forms the base of a right cone whose vertex $Z$ has coordinates $(-1, -1, 0)$ .

Find the coordinates of the centre of the circle.

[2]

Find the radius of the circle.

[2]

Find the exact volume of the cone.

[4]

Question 9

HLPaper 2

The points $A(5, -2, 5)$ , $B(5, 4, -1)$ , $C(-1, -2, -1)$ and $D(7, -4, -3)$ are the vertices of a right-pyramid.The line $L$ passes through the point $D$ and is perpendicular to plane $\Phi$

Find the vectors ${\vec{AB}}$ and ${\vec{AC}}$ .

[2]

Show that the Cartesian equation of the plane $\Phi$ that contains the triangle $ABC$ is $-x+y+z=-2$ .

[3]

Find a vector equation of the line $L$ .

[1]

Hence determine the minimum distance, $d_{\text{min}}$ , from $D$ to $\Phi$ .

[4]

Use a vector method to show that $B\hat{A}C = 60^\circ$ .

[3]

Find the volume of right-pyramid ${ABCD}$ .

[4]

Question 10

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]