Given $$\mathbf{a} = 3\mathbf{i} - \mathbf{j}$$ find $|\mathbf{a}|$.
$\sqrt{10}$
Given $$\mathbf{b} = \begin{pmatrix}-2 \\ 4\end{pmatrix}$$ find $|\mathbf{b}|$.
$\sqrt{20}$
Given $$\mathbf{c} = 2\begin{pmatrix}2 \\ -3\end{pmatrix}$$ find $|\mathbf{c}|$.
$2\sqrt{13}$
Given $$\mathbf{d} = 3\begin{pmatrix}3 \\ 1\end{pmatrix} - 2\begin{pmatrix}2 \\ -1\end{pmatrix}$$ find $|\mathbf{d}|$.
$\sqrt{50}$
Given $$\begin{pmatrix}2 \\ -5\end{pmatrix} + 2\begin{pmatrix}3 \\ k\end{pmatrix}$$ is parallel to $\begin{pmatrix}1\\0\end{pmatrix}$ find $k$.
$\dfrac{5}{2}$
Given $$3\begin{pmatrix}-3 \\ 1\end{pmatrix} + \begin{pmatrix}k \\ 3\end{pmatrix}$$ is parallel to $\begin{pmatrix}1\\1\end{pmatrix}$ find $k$.
$15$
Given $$\begin{pmatrix}2 \\ -5\end{pmatrix} + \lambda\begin{pmatrix}3 \\ 2\end{pmatrix}$$ is parallel to the vector $\begin{pmatrix}1\\0\end{pmatrix}$ find $\lambda$.
$\dfrac{5}{2}$
Given $$\begin{pmatrix}-3 \\ 1\end{pmatrix} + \lambda\begin{pmatrix}2 \\ 3\end{pmatrix}$$ is parallel to the vector $\begin{pmatrix}1\\1\end{pmatrix}$ find $\lambda$.
$-4$
Given $$\begin{pmatrix}-1 \\ 2\end{pmatrix} + \lambda\begin{pmatrix}3 \\ -1\end{pmatrix}$$ is parallel to the vector $\begin{pmatrix}2\\1\end{pmatrix}$ find $\lambda$.
$1$
Given $$\begin{pmatrix}4 \\ -3\end{pmatrix} + \lambda\begin{pmatrix}-2 \\ 1\end{pmatrix}$$ is parallel to the vector $\begin{pmatrix}2\\-3\end{pmatrix}$ find $\lambda$.
$\dfrac{3}{2}$
Given $$\mathbf{r} = 2\mathbf{i} + 4\mathbf{j}$$ find $\mathbf{\hat{r}}$.
$\dfrac{1}{\sqrt{5}}\mathbf{i} + \dfrac{2}{\sqrt{5}}\mathbf{j}$
Given $$\mathbf{s} = 3\mathbf{i} - 3\mathbf{j}$$ find the unit vector in the direction of $\mathbf{s}$.
$\dfrac{1}{\sqrt{2}}\mathbf{i} - \dfrac{1}{\sqrt{2}}\mathbf{j}$
Find the magnitude and direction of the vector $$\mathbf{i} + \mathbf{j}$$
$\sqrt{2}$, $45^{\circ}$
Find the magnitude and direction of the vector $$3\mathbf{i} - 2\mathbf{j}$$
$\sqrt{13}$, $-33.7^{\circ}$
Find the magnitude and direction of the vector $$\begin{pmatrix}-2 \\ 1\end{pmatrix}$$
$\sqrt{5}$, $153^{\circ}$
Find the magnitude and direction of the vector $$\begin{pmatrix}-3 \\ -1\end{pmatrix}$$
$\sqrt{10}$, $-162^{\circ}$
Given $$\mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$$ find $|\mathbf{a}|$.
$\sqrt{14}$
Given $$\mathbf{b} = 2\begin{pmatrix}-2 \\ 2 \\ -1\end{pmatrix} - 3\begin{pmatrix}1 \\ -2 \\ -3\end{pmatrix}$$ find $|\mathbf{b}|$.
$\sqrt{198}$
Given $$\mathbf{c} = \begin{pmatrix}4 \\ k \\ 3\end{pmatrix}$$ and that $|\mathbf{c}| = 5\sqrt{2}$, find the possible value(s) of $k$.
$-5, 5$
Given $$\mathbf{d} = \begin{pmatrix}-1 \\ k \\ 2\end{pmatrix} + 2\begin{pmatrix}k \\ 0 \\ 3\end{pmatrix}$$ and that $|\mathbf{d}| = \sqrt{66}$, find the possible value(s) of $k$.
$-0.2, 1$
Given $$\begin{pmatrix}1 \\ 2 \\ k\end{pmatrix} + \lambda\begin{pmatrix}-1 \\ 1 \\ 2\end{pmatrix}$$ is parallel to $\begin{pmatrix}2 \\ -1 \\ 4\end{pmatrix}$ find $\lambda$ and $k$.
$\lambda = -5$ and $k = 22$
Given $$\begin{pmatrix}5 \\ -1 \\ -2\end{pmatrix} + \lambda\begin{pmatrix}-4 \\ 2 \\ k\end{pmatrix}$$ is parallel to $\begin{pmatrix}3 \\ -3 \\ 2\end{pmatrix}$ find $\lambda$ and $k$.
$\lambda = 2$ and $k = 0$
Given $$\mathbf{r} = 4\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$$ find the unit vector in the direction of $\mathbf{r}$.
$\dfrac{2}{\sqrt{6}}\mathbf{i} - \dfrac{1}{\sqrt{6}}\mathbf{j} + \dfrac{1}{\sqrt{6}}\mathbf{k}$
Given $$\mathbf{s} = \begin{pmatrix}3 \\ 6 \\ -6\end{pmatrix}$$ find $\mathbf{\hat{s}}$.
$\dfrac{1}{3}\begin{pmatrix}1 \\ 2 \\ -2\end{pmatrix}$
Given $$\mathbf{a} = \begin{pmatrix}2 \\ 0 \\ 3\end{pmatrix} + \begin{pmatrix}1 \\ -1 \\ 2\end{pmatrix}$$ $$\mathbf{b} = 2\begin{pmatrix}1 \\ -2 \\ 5\end{pmatrix} - \begin{pmatrix}-1 \\ -7 \\ 4\end{pmatrix}$$ determine if $\mathbf{a}$ and $\mathbf{b}$ are parallel.
No
Given $$\mathbf{c} = 2\begin{pmatrix}5 \\ 2 \\ -3\end{pmatrix} + 3\begin{pmatrix}-3 \\ 1 \\ -4\end{pmatrix}$$ $$\mathbf{d} = 3\begin{pmatrix}2 \\ -1 \\ 4\end{pmatrix} - 5\begin{pmatrix}1 \\ -2 \\ 6\end{pmatrix}$$ determine if $\mathbf{c}$ and $\mathbf{d}$ are parallel.
Yes