How to Find Parallel Vectors

Three parallel vectors

Two vectors are parallel if one of the vectors is a multiple of the other. Formally, this is written like this:

Rule

Parallel Vectors

\vec{a} = k \cdot \vec{b} \Leftrightarrow \vec{a} ∥ \vec{b}

The equivalence arrow tells you that if one is true, the other one is also true. Math problems will often require you to use this symbol, so you should get comfortable with how it works. Remember, the right side implies the left side, and the left side implies the right side.

Example 1

Decide whether the vectors $(12, 18)$ and $(2, 3)$ are parallel.

(12, 18) = k (2, 3)

You begin by splitting this into two equations, one for the $x$ -coordinates and one for the $y$ -coordinates.

\begin{array}{l} 12 & = k \cdot 2 & 18 & = k \cdot 3 \\ k & = 6 & k & = 6 \end{array}

Because the value for

k

are equal for both the

x

- and

y

-coordinates, you know that the vectors are parallel. If the values for

k

had not been equal, the vectors would not have been parallel.

Example 2

What does $t$ need to be in order for the vectors $(- 2, 3)$ and $(t, 7)$ to be parallel?

(- 2, 3) = k \cdot (t, 7)

We split this into two equations, one for x-coordinates and one for y-coordinates. Because we want the vectors to be parallel, we need $k$ to be the same for both coordinates. That means we can do this:

\begin{array}{l} - 2 & = k \cdot t & 3 & = k \cdot 7 \\ k & = \frac{3}{7} \\ - 2 & = \frac{3}{7} \cdot t \\ t & = - \frac{14}{3} \end{array}

If $t = - \frac{14}{3}$ , the vectors are parallel.

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