Question Type 3: Determining whether two vectors are perpendicular Exercises

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

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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ 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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

Determine whether vectors $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ 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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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 $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

For what values of $k$ is the vector $\begin{pmatrix} k^2 \\ k \\ 1 \end{pmatrix}$ perpendicular to the vector $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$?

Determine whether vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ are perpendicular.