# Let F = 2i - 6j + 10k and G = i+ Gy j + 5k. If F and G have the same unit vector, what is Gy?

Nov 19, 2016

${G}_{y} = - 3$

#### Explanation:

Given: $\overline{F} = 2 \hat{i} - 6 \hat{j} + 10 \hat{k}$.

The phrase; "If $\overline{F}$ and $\overline{G}$ have the same unit vector" allows us to write the equation:

$\overline{G} = k \left(\overline{F}\right)$

where k is a scalar multiplier.

We can find the value of the multiplier by examining the $\hat{i}$ components of the two vectors:

$\hat{i} = \left(k\right) 2 \hat{i}$

$k = \frac{1}{2}$

This is consistent with the $\hat{k}$ component of the two vectors, too:

$5 \hat{k} = \left(\frac{1}{2}\right) 10 \hat{k}$

Therefore, we apply the same scalar multiplier to the $\hat{j}$ components:

${G}_{y} \hat{j} = \left(\frac{1}{2}\right) \left(- 6 \hat{j}\right)$

${G}_{y} = - 3$