Question Type 3: Finding the angle between vectors Exercises

Find the angle between the vectors $\begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}$.

Find the angle between the vectors $\begin{pmatrix} 1 \\ 4 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$.

Let $\mathbf{u}=(\cos\alpha,\sin\alpha,0)$ and $\mathbf{v}=(1,0,1)$. Find $\alpha$ such that the angle between $\mathbf{u}$ and $\mathbf{v}$ is $45^\circ$.

Find the angle between the vector sum $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 4 \\ 0 \\ -1 \end{pmatrix}$ and the vector $\begin{pmatrix} 2 \\ 3 \\ -4 \end{pmatrix}$.

Find the possible values of $x$ such that the angle between the vectors $(x,1,2)$ and $(2,3,4)$ is $60^\circ$.

Find the angle between $\mathbf{u}=(1,-1,2)$ and $\mathbf{v}=(-2,4,-1)$.

Given vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 3 \\ x \\ 5 \end{pmatrix}$ are perpendicular, find $x$ and verify that the angle between them is $90^\circ$.

Let $\mathbf{u}=(2,-1,4)$ and $\mathbf{v}=(1,3,-2)$. Find the angle between $\mathbf{u}$ and $\mathbf{v}$.

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

Find the acute angle between the vectors $\begin{pmatrix} 5 \\ -3 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 2 \\ 4 \\ 3 \end{pmatrix}$.

Determine the angle between $\mathbf{u}=(3,-3,1)$ and $\mathbf{v}=(-1,2,2)$.

Show whether the angle between the vectors $(1,2,-1)$ and $(4,0,3)$ is acute or obtuse, and find its measure.