IB Mathematics Analysis and Approaches Question Type 3: Finding the Angle Between Vectors Exercises

Find the angle between the vectors $\left(\begin{array}{c}-2\\ 3\\ 1\end{array}\right)$ and $\left(\begin{array}{c}4\\ -1\\ 2\end{array}\right)$.

Find the angle between the vectors $\left(\begin{array}{c}1\\ 4\\ 3\end{array}\right)$ and $\left(\begin{array}{c}2\\ 1\\ 2\end{array}\right)$.

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 $\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)+\left(\begin{array}{c}4\\ 0\\ -1\end{array}\right)$ and the vector $\left(\begin{array}{c}2\\ 3\\ -4\end{array}\right)$.

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 $\mathbf{a}=\left(\begin{array}{c}1\\ 2\\ 2\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}3\\ x\\ 5\end{array}\right)$ 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 $\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$ and $\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$.

Find the acute angle between the vectors $\left(\begin{array}{c}5\\ -3\\ 1\end{array}\right)$ and $\left(\begin{array}{c}2\\ 4\\ 3\end{array}\right)$.

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.