Guided Example #4 - Vectors

Find a vector in the opposite direction of $a = (5, - 12)$ that has magnitude 6.

a. $(30, - 72)$
b. $(- 30, 72)$
c. $(\frac{- 72}{13}, \frac{30}{13})$
d. $(\frac{- 30}{13}, \frac{72}{13})$

The correct answer is D. To see why, view each of the steps below.

Step 1

The magnitude of a is given by $| | a | | = \sqrt{{(5)}^{2} + {(- 12)}^{2}} = \sqrt{25 + 144} = \sqrt{169} = 13$

Step 2

A unit vector u in the direction of a can be found evaluating a $\frac{1}{| | a | |}$
$u = a \frac{1}{| | a | |} = (5, - 12) \frac{1}{13} = (\frac{5}{13}, - \frac{12}{13})$

Multiplying u by 6 gives us a vector of magnitude 6 in the direction of a, so we will multiply u by –6 to obtain the desired vector. $- 6 u = - 6 (\frac{5}{13}, - \frac{12}{13}) = (\frac{- 30}{13}, \frac{72}{13})$

All Steps