Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither.
$$\mathbf { u } = \langle 1 , - 5 \rangle , \mathbf { v } = \langle 20,3 \rangle$$
A) orthogonal
B) parallel
C) neither

Question

Quizplus · Accepted Answer

To determine if two vectors are orthogonal, we need to compute their dot product. If the dot product is zero, then they are orthogonal. If the dot product is nonzero, then they are not orthogonal. 
$$\mathbf { u } \cdot \mathbf { v } = (1)(20) + (-5)(3) = 5$$
Since the dot product is not zero, the vectors are not orthogonal. To determine if they are parallel, we can compute the magnitude of each vector and their ratio. If the ratio is a constant, then they are parallel. If not, they are neither parallel nor orthogonal. 
$$|\mathbf { u }| = \sqrt{1^2 + (-5)^2} = \sqrt{26}$$
$$|\mathbf { v }| = \sqrt{20^2 + 3^2} = \sqrt{409}$$
$$\frac{|\mathbf { u }|}{|\mathbf { v }|} = \frac{\sqrt{26}}{\sqrt{409}}$$
Since this ratio is not a constant, the vectors are not parallel. Thus, the answer is C, neither.

Determine Whether u\mathbf { u }u And v\mathbf { v }v Are Orthogonal, Parallel, or Neither

Determine Whether $\mathbf { u }$ And $\mathbf { v }$ Are Orthogonal, Parallel, or Neither