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

AHL 3.13—Scalar and vector products Questionbank

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 $=\left[\begin{array}{c}2 \\ -1 \\ 1\end{array}\right]$ and $=\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]$ .

Find the area of the triangle formed by vectors and .

[4]

Question 2

HLPaper 2

Let $=\left[\begin{array}{c}2 \\ -2 \\ 1\end{array}\right]$ and $=\left[\begin{array}{l}0 \\ 1 \\ 3\end{array}\right]$ .

Show that the angle between the vectors is acute.

[4]

Question 3

HLPaper 1

Let $=\left[\begin{array}{c}2 \\ -2 \\ 1\end{array}\right]$ .

Find a unit vector in the same direction as .

[2]

Question 4

HLPaper 1

Let $=\left[\begin{array}{c}1 \\ 3 \\ -2\end{array}\right]$ and $=\left[\begin{array}{c}4 \\ -1 \\ x\end{array}\right]$ .

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

[4]

Question 5

HLPaper 1

Let $=\left[\begin{array}{c}2 \\ -3 \\ 1\end{array}\right],=\left[\begin{array}{c}-1 \\ 4 \\ 2\end{array}\right]$ , and $=\left[\begin{array}{c}3 \\ -2 \\ 5\end{array}\right]$ .

Determine whether the three vectors lie in the same plane.

[4]

Question 6

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 7

HLPaper 2

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

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 8

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 9

SLPaper 1

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

The line through A and B is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .

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

Question 10

HLPaper 1

A particle P moves with velocity v = $\left( {\begin{array}{*{20}{c}} { - 15} \\ 2 \\ 4 \end{array}} \right)$ in a magnetic field, B = $\left( {\begin{array}{*{20}{c}} 0 \\ d \\ 1 \end{array}} \right)$ , $d \in \mathbb{R}$ .

Given that v is perpendicular to B, find the value of $d$ .

[2]

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

Given that| F | = 14, find the value of $a$ .

[4]

Paper

Level

Question 1

HLPaper 1

Let $=\left[\begin{array}{c}2 \\ -1 \\ 1\end{array}\right]$ and $=\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]$ .

Find the area of the triangle formed by vectors and .

[4]

Question 2

HLPaper 2

Let $=\left[\begin{array}{c}2 \\ -2 \\ 1\end{array}\right]$ and $=\left[\begin{array}{l}0 \\ 1 \\ 3\end{array}\right]$ .

Show that the angle between the vectors is acute.

[4]

Question 3

HLPaper 1

Let $=\left[\begin{array}{c}2 \\ -2 \\ 1\end{array}\right]$ .

Find a unit vector in the same direction as .

[2]

Question 4

HLPaper 1

Let $=\left[\begin{array}{c}1 \\ 3 \\ -2\end{array}\right]$ and $=\left[\begin{array}{c}4 \\ -1 \\ x\end{array}\right]$ .

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

[4]

Question 5

HLPaper 1

Let $=\left[\begin{array}{c}2 \\ -3 \\ 1\end{array}\right],=\left[\begin{array}{c}-1 \\ 4 \\ 2\end{array}\right]$ , and $=\left[\begin{array}{c}3 \\ -2 \\ 5\end{array}\right]$ .

Determine whether the three vectors lie in the same plane.

[4]

Question 6

HLPaper 2

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

[4]

Question 7

HLPaper 2

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

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 8

HLPaper 2

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

[5]

Question 9

SLPaper 1

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

The line through A and B is perpendicular to the vector 2i $+$ nk. Find the value of $n$ .

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

Question 10

HLPaper 1

Given that v is perpendicular to B, find the value of $d$ .

[2]

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

Given that| F | = 14, find the value of $a$ .

[4]