Vectors and Transformations

Find $|\vec{h}|$ where $\vec{h} = \binom{7}{7}$.

What is the magnitude of the vector $\binom{-3}{0}$?

If $\vec{x} = \binom{2}{3}$, find $\vec{x} + \vec{x} + \vec{x}$.

Find the magnitude of $\vec{g} = 15\mathbf{i} + 20\mathbf{j}$.

If $\vec{a} = \binom{1}{2}$ and $\vec{b} = \binom{2}{1}$, find the magnitude $|10\vec{a} + 10\vec{b}|$.

Find the magnitude of the vector $\binom{\sqrt{2}}{\sqrt{7}}$.

Given $\vec{m} = \binom{6}{-8}$ and $\vec{n} = \binom{-3}{4}$, find the magnitude $|\vec{m} + 2\vec{n}|$.

If $|\binom{x}{y}| = 5$ and $|\binom{x}{2y}| = \sqrt{73}$, find $|x|$.

Find the magnitude of $\vec{a} - \vec{b}$ if $\vec{a} = \binom{4}{10}$ and $\vec{b} = \binom{1}{6}$.

The vector $\vec{v} = \binom{m+1}{3}$ has magnitude 5. Find the sum of possible values of $m$.

Find the difference $\begin{pmatrix} 9 \\ 9 \end{pmatrix} - \begin{pmatrix} 10 \\ 10 \end{pmatrix}$.

Calculate the sum: $\begin{pmatrix} 3 \\ 4 \end{pmatrix} +$\begin{pmatrix} 1 \ 1 \end{pmatrix}$+ \begin{pmatrix} -2 \\ -2 \end{pmatrix}$.

Calculate $\begin{pmatrix} 0 \\ 0 \end{pmatrix} - \begin{pmatrix} 4 \\ 7 \end{pmatrix}$.

Find the result of $\begin{pmatrix} 15 \\ 5 \end{pmatrix} - \begin{pmatrix} 10 \\ 2 \end{pmatrix}$.

If 5\mathbf{x} - $$\begin{pmatrix} 2 \\ 3 \end{pmatrix}$$ = 2\mathbf{x} + \begin{pmatrix} 7 \\ 12 \end{pmatrix}, find the vector $\mathbf{x}$.

In a triangle $ABC$, $\vec{AB} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$ and $\vec{AC} = \begin{pmatrix} 6 \\ 1 \end{pmatrix}$. Find $\vec{BC}$.

Calculate 4$$\begin{pmatrix} -1 \\ 2 \end{pmatrix}$$ - 3\begin{pmatrix} -2 \\ -3 \end{pmatrix}.

A path is defined by $O \to A \to B \to C$. $\vec{OA} = \mathbf{a}$, $\vec{AB} = 2\mathbf{a} + \mathbf{b}$, and $\vec{BC} = \mathbf{b} - \mathbf{a}$. Find $\vec{OC}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

In triangle $ABC$, $\vec{AB} = \mathbf{a}$ and $\vec{AC} = \mathbf{b}$. $P$ is on $BC$ such that $BP:PC = 1:2$. Find $\vec{AP}$.

Given $\vec{XY} = \mathbf{k}$. If $\vec{XZ} = 3\mathbf{k}$ and $\vec{XW} = -2\mathbf{k}$, find $\vec{ZW}$ in terms of $\mathbf{k}$.

Point $U(3, 3)$ is reflected in the line $y = -x$. Find $U'$.

Translate the point $(0, -4)$ by the vector $\binom{-2}{8}$.

The point $(5, 2)$ is rotated $180^\circ$ about the origin. Where is it now?

An enlargement with center $(0,0)$ and scale factor $k=3$ maps $(x, y)$ to:

A point $H(2, 4)$ is rotated $90^\circ$ anticlockwise about the point $(1, 1)$. What is the image $H'$?

A translation followed by another translation is always equivalent to a single:

The point $(-3, -3)$ is reflected in the line $y = x$. The result is:

Given the translation $T: (x, y) \to (x+a, y+b)$, if $T$ followed by a reflection in $y=0$ maps $(1, 2)$ to $(3, -5)$, find $a$ and $b$.

A reflection in the line $y = x$ is followed by a reflection in the $x$-axis. This is equivalent to:

The point $(-2, 5)$ is rotated $270^{\circ}$ clockwise about the origin. What are the coordinates of the image?