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AHL 3.13—Scalar and vector products - IB Questionbank | RevisionDojo

Practice AHL 3.13—Scalar and vector products with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}$ .

Find the area of the triangle formed by vectors $\mathbf{a}$ and $\mathbf{b}$ .

[4]

Question 2

HLPaper 2

Two lines in vector form are given in a 3-dimensional space.

The line $L_1$ is given by $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and the line $L_2$ is given by $\mathbf{r} = \begin{pmatrix} 4 \\ 0 \\ 1 \end{pmatrix} + s \begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix}$ . Find the angle between the two lines.

[4]

Question 3

HLPaper 2

Consider two vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ .

Tip:

Remember to show all steps clearly when calculating the dot product and magnitudes

Find the scalar product (dot product) of vectors $\mathbf{a}$ and $\mathbf{b}$ .

[2]

Calculate the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ .

[3]

Question 4

HLPaper 2

Consider two lines in three-dimensional space given by their parametric equations.

Line 1 is given by the parametric equations $x = 1 + 2t$ , $y = -1 + 3t$ , $z = 4t$ . Line 2 is given by the parametric equations $x = 3 + s$ , $y = 2 - s$ , $z = 5 + 2s$ . Find the shortest distance between these two lines by finding the perpendicular distance.

[5]

Question 5

HLPaper 2

Let $\mathbf{u} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}$ .

Show that the angle between the vectors $\mathbf{u}$ and $\mathbf{v}$ is acute.

[4]

Question 6

HLPaper 1

Let $\boldsymbol{v} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ .

Find a unit vector in the same direction as $\boldsymbol{v}$ .

[2]

Question 7

SLPaper 1

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

The line through A and B is perpendicular to the vector $2\mathbf{i} + n\mathbf{k}$ . Find the value of $n$ .

[3]

Find a vector equation of the line that passes through A and B.

[4]

Question 8

HLPaper 1

A particle P moves with velocity $\boldsymbol{v} = \begin{pmatrix} -15 \\ 2 \\ 4 \end{pmatrix}$ in a magnetic field, $\boldsymbol{B} = \begin{pmatrix} 0 \\ d \\ 1 \end{pmatrix}$, $d \in \mathbb{R}$.

Given that $\boldsymbol{v}$ is perpendicular to $\boldsymbol{B}$, find the value of $d$.

[2]

The force, $\boldsymbol{F}$, produced by P moving in the magnetic field is given by the vector equation $\boldsymbol{F} = a \boldsymbol{v} \times \boldsymbol{B}$, $a \in \mathbb{R}^+$.

Given that $|\boldsymbol{F}| = 14$, find the value of $a$.

[4]

Question 9

HLPaper 1

Let $\vec{a} = \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \\ -1 \\ x \end{pmatrix}$ .

Find the value of $x$ such that the angle between $\vec{a}$ and $\vec{b}$ is $90^{\circ}$ .

[4]

Question 10

HLPaper 1

Let $\vec{a} = \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}$ , $\vec{b} = \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}$ , and $\vec{c} = \begin{bmatrix} 3 \\ -2 \\ 5 \end{bmatrix}$ .

Determine whether the three vectors lie in the same plane.

[4]

Paper

Level

Question 1

HLPaper 1

Let $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}$ .

Find the area of the triangle formed by vectors $\mathbf{a}$ and $\mathbf{b}$ .

[4]

Question 2

HLPaper 2

Two lines in vector form are given in a 3-dimensional space.

[4]

Question 3

HLPaper 2

Consider two vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ .

Tip:

Remember to show all steps clearly when calculating the dot product and magnitudes

Find the scalar product (dot product) of vectors $\mathbf{a}$ and $\mathbf{b}$ .

[2]

Calculate the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ .

[3]

Question 4

HLPaper 2

Consider two lines in three-dimensional space given by their parametric equations.

[5]

Question 5

HLPaper 2

Let $\mathbf{u} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}$ .

Show that the angle between the vectors $\mathbf{u}$ and $\mathbf{v}$ is acute.

[4]

Question 6

HLPaper 1

Let $\boldsymbol{v} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ .

Find a unit vector in the same direction as $\boldsymbol{v}$ .

[2]

Question 7

SLPaper 1

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

The line through A and B is perpendicular to the vector $2\mathbf{i} + n\mathbf{k}$ . Find the value of $n$ .

[3]

Find a vector equation of the line that passes through A and B.

[4]

Question 8

HLPaper 1

A particle P moves with velocity $\boldsymbol{v} = \begin{pmatrix} -15 \\ 2 \\ 4 \end{pmatrix}$ in a magnetic field, $\boldsymbol{B} = \begin{pmatrix} 0 \\ d \\ 1 \end{pmatrix}$, $d \in \mathbb{R}$.

Given that $\boldsymbol{v}$ is perpendicular to $\boldsymbol{B}$, find the value of $d$.

[2]

The force, $\boldsymbol{F}$, produced by P moving in the magnetic field is given by the vector equation $\boldsymbol{F} = a \boldsymbol{v} \times \boldsymbol{B}$, $a \in \mathbb{R}^+$.

Given that $|\boldsymbol{F}| = 14$, find the value of $a$.

[4]

Question 9

HLPaper 1

Let $\vec{a} = \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \\ -1 \\ x \end{pmatrix}$ .

Find the value of $x$ such that the angle between $\vec{a}$ and $\vec{b}$ is $90^{\circ}$ .

[4]

Question 10

HLPaper 1

Let $\vec{a} = \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}$ , $\vec{b} = \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}$ , and $\vec{c} = \begin{bmatrix} 3 \\ -2 \\ 5 \end{bmatrix}$ .

Determine whether the three vectors lie in the same plane.

[4]

Paper

Level

Question 1

HLPaper 1

Let $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}$ .

Find the area of the triangle formed by vectors $\mathbf{a}$ and $\mathbf{b}$ .

[4]

Question 2

HLPaper 2

Two lines in vector form are given in a 3-dimensional space.

[4]

Question 3

HLPaper 2

Consider two vectors $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ .

Tip:

Remember to show all steps clearly when calculating the dot product and magnitudes

Find the scalar product (dot product) of vectors $\mathbf{a}$ and $\mathbf{b}$ .

[2]

Calculate the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ .

[3]

Question 4

HLPaper 2

Consider two lines in three-dimensional space given by their parametric equations.

[5]

Question 5

HLPaper 2

Let $\mathbf{u} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ 3 \end{pmatrix}$ .

Show that the angle between the vectors $\mathbf{u}$ and $\mathbf{v}$ is acute.

[4]

Question 6

HLPaper 1

Let $\boldsymbol{v} = \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$ .

Find a unit vector in the same direction as $\boldsymbol{v}$ .

[2]

Question 7

SLPaper 1

The position vectors of points A and B are $\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $7\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$ respectively.

The line through A and B is perpendicular to the vector $2\mathbf{i} + n\mathbf{k}$ . Find the value of $n$ .

[3]

Find a vector equation of the line that passes through A and B.

[4]

Question 8

HLPaper 1

A particle P moves with velocity $\boldsymbol{v} = \begin{pmatrix} -15 \\ 2 \\ 4 \end{pmatrix}$ in a magnetic field, $\boldsymbol{B} = \begin{pmatrix} 0 \\ d \\ 1 \end{pmatrix}$, $d \in \mathbb{R}$.

Given that $\boldsymbol{v}$ is perpendicular to $\boldsymbol{B}$, find the value of $d$.

[2]

The force, $\boldsymbol{F}$, produced by P moving in the magnetic field is given by the vector equation $\boldsymbol{F} = a \boldsymbol{v} \times \boldsymbol{B}$, $a \in \mathbb{R}^+$.

Given that $|\boldsymbol{F}| = 14$, find the value of $a$.

[4]

Question 9

HLPaper 1

Let $\vec{a} = \begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 4 \\ -1 \\ x \end{pmatrix}$ .

Find the value of $x$ such that the angle between $\vec{a}$ and $\vec{b}$ is $90^{\circ}$ .

[4]

Question 10

HLPaper 1

Let $\vec{a} = \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}$ , $\vec{b} = \begin{bmatrix} -1 \\ 4 \\ 2 \end{bmatrix}$ , and $\vec{c} = \begin{bmatrix} 3 \\ -2 \\ 5 \end{bmatrix}$ .

Determine whether the three vectors lie in the same plane.

[4]

AHL 3.13—Scalar and vector products Questionbank