Vectors and Transformations

Find $|\mathbf{m}|$ if $\mathbf{m} = \begin{pmatrix} 24 \\ 10 \end{pmatrix}$.

What is the magnitude of the vector $\vec{k} = \begin{pmatrix} 8 \\ 2 \end{pmatrix}$?

Determine the magnitude of the vector $\mathbf{n} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$.

Find the magnitude of the vector $\mathbf{x} = 3\mathbf{i} + 6\mathbf{j}$.

Given $\mathbf{a} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 5 \\ 12 \end{pmatrix}$, find the sum of their magnitudes, $|\mathbf{a}| + |\mathbf{b}|$.

If $\mathbf{a} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 9 \end{pmatrix}$, calculate the magnitude of the resultant vector $\mathbf{a} + \mathbf{b}$.

What is the magnitude of vector $\mathbf{q} = \begin{pmatrix} 36 \\ 15 \end{pmatrix}$?

If $\mathbf{v} = \begin{pmatrix} -12 \\ 5 \end{pmatrix}$, find the magnitude of the vector $0.5\mathbf{v}$.

Given $\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$, calculate the magnitude of the vector $3\mathbf{a} - \mathbf{a}$.

If $\vec{LM} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\vec{MN} = \begin{pmatrix} 4 \\ 7 \end{pmatrix}$, find the exact magnitude of vector $\vec{LN}$.

If $\begin{pmatrix} x \\ y \end{pmatrix} +$\begin{pmatrix} 2 \ 3 \end{pmatrix}$= \begin{pmatrix} 5 \\ 5 \end{pmatrix}$, find the values of $x$ and $y$.

Find the result of $\vec{a} + \vec{b}$ where $\vec{a} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -6 \\ -2 \end{pmatrix}$.

If $\vec{PQ} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$, what is $3\vec{PQ}$?

The resultant of the vectors $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ is:

If $\vec{v} = \begin{pmatrix} 5 \\ -12 \end{pmatrix}$, find the resultant vector of $3\vec{v} + \begin{pmatrix} 2 \\ 6 \end{pmatrix}$.

Given $\vec{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -2 \\ 6 \end{pmatrix}$, find $\frac{1}{2}(\vec{a} + \vec{b})$.

In $\triangle OAB$, $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$. Point $M$ is the midpoint of $OA$. Find $\vec{MB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Given $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$. If $C$ is a point such that $\vec{OC} = 2\mathbf{a} - 3\mathbf{b}$, express $\vec{BC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

If $\vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$, find $k$ such that $\vec{a} + k\vec{b}$ is parallel to $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$.

In a parallelogram $ABCD$, $\vec{AB} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}$ and $\vec{BC} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$. Find the vector $\vec{AC}$.

Point $P(2, 4)$ is translated by the vector $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$. What are the coordinates of the image $P'$?

Point $O(2, 3)$ is enlarged from the origin $(0, 0)$ with a scale factor of $-1$. What are the coordinates of the image $O'$?

Point $N(-2, -2)$ is reflected in the line $y = x$. What are the coordinates of the image $N'$?

Point $M(1, 2)$ is rotated $180^\circ$ about the origin $(0, 0)$. What are the coordinates of the image $M'$?

If the area of a shape increases from $10\text{ cm}^2$ to $40\text{ cm}^2$ after an enlargement, what is the positive scale factor?

An enlargement with center $(2, 2)$ and scale factor $2$ maps the point $(1, 1)$ onto which point?

The point $(1, 2)$ is reflected in the line $y = -x$. What is the image?

What is the result of rotating the point $(3, 0)$ by $45^\circ$ anti-clockwise about the origin?

A triangle has area $A$. It is enlarged by scale factor $k$, then the result is enlarged by scale factor $m$. What is the area of the final image?

Determine the matrix for a shear with $y$-axis invariant that maps $(1, 0)$ to $(1, -3)$.