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AHL 3.13—Scalar (dot) product - IB Questionbank | RevisionDojo

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

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 4

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 5

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 6

HLPaper 1

The cube alongside has edges of length 2 cm . Find the measure of

$\angle A B S$

[4]

$\angle R B P$

[4]

Question 7

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 8

HLPaper 1

Find the cosine of the angle between the two vectors

$\binom{3}{4}$ and $\binom{-2}{1}$

[4]

Question 9

HLPaper 1

Find any two non - vectors which are not parallel, but which are both perpendicular to

$\left(\begin{array}{c}1 \\ 2 \\ -1\end{array}\right)$ .

[7]

Question 10

HLPaper 2

Given a parallelogram $O P Q R$ in which $\overrightarrow{O P}=\binom{7}{3}$ and $\overrightarrow{O Q}=\binom{10}{1}$

Find the vector $\overrightarrow{O R}$

[2]

Use the scalar product of two vectors to show that $\cos \angle O P Q=\frac{-15}{\sqrt{754}}$

[5]

Explain why $\cos \angle P Q R=-\cos \angle O P Q$

[2]

Hence show that $\sin \angle P Q R=\frac{23}{\sqrt{754}}$

[3]

AHL 3.13—Scalar (dot) product Questionbank