Question

Determine whether the vectors u and v are parallel, orthogonal, or neither.

Answer 1

If the vectors are parallel:
→       →
u = k · v
8 = 10 k
4 = 7  k.
If they are orthogonal:
→  →
u ·  v = 0
( 10 · 8 ) + ( 7 · 4 ) 0 80 + 28 = 108 ≠ 0
Answer: the vectors u and v are neither parallel nor orthogonal.

Answer 2

Answer:

The vectors are neither parallel nor orthogonal.

Step-by-step explanation:

We are given the vectors as,

u = <8,4> and v = <10,7>

So, their dot product is given by,

u · v = <8,4> · <10,7> = 8×10 + 4×7 = 80 + 28 = 108 ≠ 0

As, we know, Two vectors are orthogonal if their dot product is 0.

Since, u · v = 108 ≠ 0

Thus, they are not orthogonal.

Also, Two vectors are parallel if angle between them is 0° or 180°.

So, we will find the angle between u and v.

As, ║u║ = $\sqrt{8^{2}+4^{2}}$ = $\sqrt{64+16}$ = $\sqrt{80}$ = 8.94

As, ║v║ = $\sqrt{10^{2}+7^{2}}$ = $\sqrt{100+49}$ = $\sqrt{149}$ = 12.2

So, we have,

$\theta = \cos^{-1} \frac{u\cdot v}{\left \| u \right \|\times \left \| v \right \|}$

i.e. $\theta = \cos^{-1} \frac{108}{8.94\times 12.2|}$

i.e. $\theta = \cos^{-1} \frac{108}{109.068}$

i.e. $\theta = \cos^{-1} 0.9902$

i.e. $\theta = 8.03$

Thus, θ ≈ 8.03°, which is not equal to 0° or 180°.

Then, the vectors are not parallel.

Hence, we see that the vectors are neither parallel nor orthogonal.