Question Type 4: Determining if combinations of the specific vectors are parallel, perpendicular or neither Exercises

Determine whether $4\mathbf{a} - \mathbf{b} + \mathbf{c}$ is orthogonal to $\mathbf{a} + 3\mathbf{b} - 2\mathbf{c}$, 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 the vector $2\mathbf{a} - 3\mathbf{b}$ is parallel, perpendicular, or neither to the vector $\mathbf{a} + \mathbf{b}$, where $\mathbf{a}=(1,3,4)$ and $\mathbf{b}=(1,0,1)$.

Are the vectors $\mathbf{a} - 3\mathbf{b} + \mathbf{c}$ and $-2\mathbf{a} + 6\mathbf{b} - 2\mathbf{c}$ parallel, perpendicular, or neither? Use $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

Determine whether $3(\mathbf{a} - \mathbf{b}) + 4\mathbf{c}$ is parallel, perpendicular, or neither to $-6(\mathbf{a} - \mathbf{b}) - 8\mathbf{c}$, 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 the vectors $\boldsymbol{a} + 2\boldsymbol{c}$ and $4\boldsymbol{b} - \boldsymbol{c}$ are parallel, perpendicular, or neither, where $\boldsymbol{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\boldsymbol{b}=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, and $\boldsymbol{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

Determine whether the vector $3\mathbf{a} - \mathbf{b} + \mathbf{c}$ is perpendicular to $\mathbf{a} - 2\mathbf{c}$, 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 the vectors $2\mathbf{a} + 3\mathbf{b}$ and $-4\mathbf{a} - 6\mathbf{b}$ are parallel, perpendicular, or neither, where $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$.

Determine whether the vector $\mathbf{a} + (\mathbf{b} + \mathbf{c})$ is perpendicular, parallel, or neither to the vector $\mathbf{a} - (\mathbf{b} - \mathbf{c})$, given $\mathbf{a}=\begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b}=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, and $\mathbf{c}=\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

Determine if the vectors $\boldsymbol{a} + 2\boldsymbol{b} - \boldsymbol{c}$ and $2\boldsymbol{a} - 4\boldsymbol{b} + 2\boldsymbol{c}$ are parallel, perpendicular, or neither, where $\boldsymbol{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\boldsymbol{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ and $\boldsymbol{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

Determine if the vector $5\mathbf{a} - 2\mathbf{c}$ is perpendicular to $2\mathbf{a} + \mathbf{b}$, given $\mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$.

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

Determine whether the vectors $\mathbf{u} = \mathbf{a} - \mathbf{b} + 2\mathbf{c}$ and $\mathbf{v} = 3\mathbf{a} + 6\mathbf{b} - 4\mathbf{c}$ are parallel, perpendicular, or neither.

Given $\boldsymbol{a} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$, $\boldsymbol{b} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, and $\boldsymbol{c} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, determine whether the vector $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}$ is parallel to $\boldsymbol{b} + 2\boldsymbol{c}$.