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Question 1

The following question concerns the scalar product of two vectors and the determination of the angle between them.

Given two vectors $\mathbf{a}$ and $\mathbf{b}$ satisfy $\mathbf{a} \cdot \mathbf{b} = 5$ , $|\mathbf{a}| = 3$ , and $|\mathbf{b}| = 4$ , find the angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ .

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Question 2

Find the value of $t$ such that the angle between the vectors $\begin{pmatrix} 1 \\ 0 \\ t \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ is $60^\circ$ .

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Question 3

The position vectors of points $A$ and $B$ are given. The angle between the vectors $\vec{OA}$ and $\vec{OB}$ is to be calculated using the scalar product formula.

Find the angle between $\vec{OA}$ and $\vec{OB}$ , where $A=(2, 0, 1)$ and $B=(1, 2, 2)$ .

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Question 4

Find the angle between the vectors $\begin{pmatrix} 4 \\ 1 \\ -4 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$ .

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Question 5

Vectors, Dot Product, Magnitudes

Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $|\mathbf{a}|=4$ , $|\mathbf{b}|=3$ and $\mathbf{a}\cdot\mathbf{b}=6$ .

Find the angle between the vectors $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ .

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Question 6

Calculate the angle between the vectors $\begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ .

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Question 7

Calculate the angle between the vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 4 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$ .

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Question 8

Determine the angle between the vectors $\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ .

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Question 9

Vectors - Angle between two vectors

Calculate the angle between the vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}$ .

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Question 10

Determine the angle between the vectors $\mathbf{a} = \begin{pmatrix} -1 \\ 2 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ -6 \\ 1 \end{pmatrix}$ .

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Question 11

Determine the angle between the vectors $\mathbf{u} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$ .

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Question 12

Find the angle between the vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ .

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Question 13

Find the value of $t$ such that the vectors $\begin{pmatrix} t \\ 1 \\ 2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ t \\ 3 \end{pmatrix}$ are perpendicular.

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Question Type 2: Finding the angle between two different vectors Exercises