Question Type 3: Doing simple calculations with vectors Exercises

Compute $3v + w$ where $v = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $w = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.

Given $\mathbf{v} = (4, 6, 1)$ and $\mathbf{w} = (3, 4, 3)$, compute $-\mathbf{v} + 4\mathbf{w}$.

Determine a unit vector perpendicular to both $\mathbf{v} = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.

Compute the distance between the points represented by $v=(4,6,1)$ and $2w=2(3,4,3)$.

Determine the magnitude of the vector $-\mathbf{v} + 4\mathbf{w}$ given $\mathbf{v} = (4,6,1)$ and $\mathbf{w} = (3,4,3)$.

Calculate the dot product $\mathbf{v} \cdot \mathbf{w}$ given $\mathbf{v} = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.

Find the projection of $\mathbf{v}=(4, 6, 1)$ onto $\mathbf{w}=(3, 4, 3)$.

Compute $2\mathbf{v} - 3\mathbf{w}$ for $\mathbf{v} = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.

Calculate the angle $\theta$ between the vectors $\mathbf{v} = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.

Find the cross product $\mathbf{v} \times \mathbf{w}$ for $\mathbf{v} = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.

Find $\mathbf{v} - 2\mathbf{w}$ for $\mathbf{v} = (4, 6, 1)$ and $\mathbf{w} = (3, 4, 3)$.

Find the unit vector in the direction of $-\mathbf{v} + 4\mathbf{w}$ for $\mathbf{v} = \begin{pmatrix} 4 \\ 6 \\ 1 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix}$.