Question

select all of the vectors below that are parallel to

$$\begin{pmatrix}-4\\\\2\\end{pmatrix}$$

Explanation:

Step1: Define parallel vector condition

A vector $\vec{v}$ is parallel to

$$\begin{pmatrix}-4\\2\end{pmatrix}$$

if $\vec{v} = k

$$\begin{pmatrix}-4\\2\end{pmatrix}$$

$ for some scalar $k$, meaning the ratio of components is $\frac{y}{x} = \frac{2}{-4} = -\frac{1}{2}$.

Step2: Find components of each vector

• $\vec{a}$:

$$\begin{pmatrix}-8\\4\end{pmatrix}$$

, ratio $\frac{4}{-8}=-\frac{1}{2}$

• $\vec{b}$:

$$\begin{pmatrix}-2\\-1\end{pmatrix}$$

, ratio $\frac{-1}{-2}=\frac{1}{2}$

• $\vec{c}$:

$$\begin{pmatrix}4\\-2\end{pmatrix}$$

, ratio $\frac{-2}{4}=-\frac{1}{2}$

• $\vec{d}$:

$$\begin{pmatrix}6\\-3\end{pmatrix}$$

, ratio $\frac{-3}{6}=-\frac{1}{2}$

• $\vec{e}$:

$$\begin{pmatrix}-4\\-2\end{pmatrix}$$

, ratio $\frac{-2}{-4}=\frac{1}{2}$

Step3: Match ratio to target

Vectors with ratio $-\frac{1}{2}$ are parallel.

Answer:

$\vec{a}$, $\vec{c}$, $\vec{d}$