Vectors and Transformations

Find the result of $\vec{XY} + \vec{YX}$.

Simplify $10\mathbf{i} - (2\mathbf{i} + \mathbf{j})$.

Scale the vector $\begin{pmatrix} -5 \\ -10 \end{pmatrix}$ by the factor $-\frac{1}{5}$.

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

Given $\mathbf{a} = \binom{3}{4}$, $\mathbf{b} = \binom{1}{-2}$, and $\mathbf{c} = \binom{0}{5}$, calculate the vector $2\mathbf{a} - \mathbf{b} + 3\mathbf{c}$.

The point $C$ divides the line segment $AB$ such that $AC:CB = 2:3$. If $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$, find $\vec{OC}$.

If $\vec{GH} = \binom{-2}{4}$, what is the value of the scaled vector $-0.5\vec{GH}$?

If $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$, and point $C$ is chosen such that $3\vec{AC} = 2\vec{AB}$, find $\vec{OC}$.

In a regular hexagon $ABCDEF$, $\vec{AB} = \mathbf{a}$ and $\vec{BC} = \mathbf{b}$. Find $\vec{BE}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Given $\vec{u} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ and $\vec{v} = \begin{pmatrix} 12 \\ 5 \end{pmatrix}$. Find the value of $|\vec{u}| + |\vec{v}| - |\vec{u} + \vec{v}|$.

Determine the magnitude of the vector $\mathbf{z} = 11\mathbf{i} + 60\mathbf{j}$.

Calculate the magnitude of $\mathbf{w} = \begin{pmatrix} -12 \\ -35 \end{pmatrix}$.

Find the magnitude of the vector $\vec{MN}$ where $M(-1, -1)$ and $N(3, 2)$.

Given points $X(0, 5)$ and $Y(5, 17)$, find the magnitude of the displacement vector $\vec{XY}$.

Determine | $$\begin{pmatrix} -2 \\ -2 \\ -1 \end{pmatrix}$$ |.

Find $x$ such that the magnitude of $\begin{pmatrix} x \\ 8 \\ 0 \end{pmatrix}$ is 10, given $x > 0$.

Calculate the magnitude of the vector \mathbf{g} = $$\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$$ - \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}.

If the vector $\mathbf{w} = \begin{pmatrix} 2^x \\ 2^x \\ 2^x \end{pmatrix}$ has magnitude $\sqrt{3}$, what is the value of $x$?

Find the magnitude of the vector $\mathbf{m} = \begin{pmatrix} \sqrt{7} \\ \sqrt{8} \\ \sqrt{10} \end{pmatrix}$.

Determine the magnitude of the vector $\vec{v} = (10 - 2)\mathbf{i} + (5 - 1)\mathbf{j} + (10 - 6)\mathbf{k}$.

Point $(4, 0)$ is rotated $90^\circ$ clockwise about the origin. The image is:

Calculate the magnitude of the vector $\binom{1}{\sqrt{3}}$.

What vector represents the translation from $(-2, -2)$ to $(2, 2)$?

Reflect the point $(5, 5)$ in the line $x = 0$.

If the translation $T = \binom{4}{-2}$ is applied twice to the point $(1, 1)$, what is the final image?

The point $Z(a, b)$ is reflected in the $y$-axis and then in the $x$-axis. This is equivalent to which single transformation?

Which vector is perpendicular to $\binom{4}{3}$ and has the same magnitude?

What is the matrix for a rotation of $270^\circ$ anti-clockwise about the origin?

The vector $\vec{v} = \binom{3}{4}$ is reflected in the line $x=0$. Find the new vector.

Find the values of $a$ and $b$ if $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$\begin{pmatrix} 1 \ 2 \end{pmatrix}$= \begin{pmatrix} 3 \\ -4 \end{pmatrix}$.

If a shear transformation is given by the matrix $\begin{pmatrix} 1 & -3 \\ 0 & 1 \end{pmatrix}$, what is the shear factor?

An enlargement with center $(0,0)$ and scale factor $k$ maps $(1, 0)$ to:

A shear with $x$-axis invariant maps $(2, 1)$ to $(5, 1)$. What is the image of $(2, 2)$ under the same shear?

The point $(1, 1)$ is enlarged by SF $3$ with center $(1, 1)$. What is the image?

If an enlargement maps $P(x, y)$ to $P'(kx, ky)$, what is the center of enlargement?

A shape has an area of $5$. It is enlarged by a factor of $k=2$, then sheared with factor $k=3$. What is the final area?

The matrix $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ is multiplied by $\begin{pmatrix} 1 & m \\ 0 & 1 \end{pmatrix}$. The resulting matrix represents:

If an enlargement maps the segment from $(1, 2)$ to $(3, 4)$ to a segment of length $10\sqrt{2}$, find the absolute value of the scale factor $k$.

Find the image of the point $(3, 2)$ under an enlargement of scale factor 2 centered at $(1, 1)$, followed by a shear of factor 3 with the $x$-axis invariant.

A composite transformation is given by $M = \begin{pmatrix} 2 & 10 \\ 0 & 2 \end{pmatrix}$. If a shape has an initial area of $A$, what is the area of the image?