IB Mathematics Applications & Interpretation Question Type 3: Determining Whether Two Vectors Are Perpendicular Exercises

Find the values of $k$ for which the vector $\mathbf{u}=\left(\begin{array}{c}{k}^2\\ k\\ 1\end{array}\right)$ is perpendicular to the vector $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Determine whether the vector $\mathbf{a}+\mathbf{b}$ is perpendicular to $\mathbf{c}$, where $\mathbf{a}=(1,3,4)$, $\mathbf{b}=(1,0,1)$, and $\mathbf{c}=(0,1,0)$.

Determine whether vectors $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$ are perpendicular.

For what real values of $k$ is the vector $(k^2,k,1)$ perpendicular to the vector $\mathbf{b}=(1,0,1)$?

Determine whether the vector $2\mathbf{b}+\mathbf{c}$ is perpendicular to $\mathbf{a}$, where $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$, and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Determine whether vectors $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$ and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$ are perpendicular.

Check if the vector $\mathbf{a}-2\mathbf{c}$ is perpendicular to $\mathbf{b}$, where $\mathbf{a}=(1,3,4)$, $\mathbf{b}=(1,0,1)$, and $\mathbf{c}=(0,1,0)$.

Find the scalar $t$ such that $\mathbf{a}+t\mathbf{b}$ is perpendicular to $\mathbf{c}$, given that $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$, and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

Determine the values of $k$ for which the vector $(k^2,k,1)$ is parallel to $(1,-2,1)$.

Find the scalar $s$ such that $s\mathbf{a}+\mathbf{c}$ is perpendicular to $\mathbf{b}$, given that $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$, $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$, and $\mathbf{c}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$.

For what values of $k$ is the vector $\left(\begin{array}{c}{k}^2\\ k\\ 1\end{array}\right)$ perpendicular to the vector $\left(\begin{array}{c}1\\ -2\\ 1\end{array}\right)$?

Determine whether vectors $\mathbf{a}=\left(\begin{array}{c}1\\ 3\\ 4\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)$ are perpendicular.