IB Mathematics Applications & Interpretation Question Type 1: Finding the Dot Product of Two Different Vectors Exercises

Two unit vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $\mathbf{u}\cdot \mathbf{v}=\frac{1}{2}$. Find the angle between $\mathbf{u}$ and $\mathbf{v}$.

In a parallelogram, the diagonals are given by $\mathbf{a}+\mathbf{b}$ and $\mathbf{a}-\mathbf{b}$ for $\mathbf{a}=\left(\begin{array}{c}2\\ 1\\ 0\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$. Find the angle between the diagonals.

Determine the value of $k$ such that the vectors $(k,2,3)$ and $(1,k,4)$ are perpendicular.

Determine whether the vectors $\mathbf{a}=(1,0,1)$ and $\mathbf{c}=(5,6,1)$ are perpendicular.

Show that the angle between the vectors $(1,1,1)$ and $(1,0,0)$ is acute and find its measure.

Given the vectors $\mathbf{a}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}3\\ 2\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}5\\ 6\\ 1\end{array}\right)$, find the value of $\mathbf{b}\cdot (\mathbf{a}-\mathbf{c})$.

Calculate the value of $\mathbf{b}\cdot \mathbf{c}-2\mathbf{a}\cdot \mathbf{c}$ for $\mathbf{a}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}3\\ 2\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}5\\ 6\\ 1\end{array}\right)$.

Verify the distributive property $\mathbf{a}\cdot (\mathbf{b}+2\mathbf{c})=\mathbf{a}\cdot \mathbf{b}+2\mathbf{a}\cdot \mathbf{c}$ for $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 0\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 2\end{array}\right)$ and compute the resulting value.

Find the angle between the vectors $\mathbf{p}=(3,-1,2)$ and $\mathbf{q}=(1,2,-2)$.

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

Compute the dot product $\mathbf{a}\cdot (\mathbf{b}+2\mathbf{c})$ for $\mathbf{a}=(1,0,1)$, $\mathbf{b}=(3,2,1)$ and $\mathbf{c}=(5,6,1)$.

Find all real values of $t$ such that the vectors $(1,t,t^2)$ and $(2,-1,1)$ are orthogonal.