Find value a when the angle between vectors U=(2,a) and V = (-6;8) is pie/4​

Step-by-step explanation:

The formula to find the angle between 2 vectors is

which is the dot product of u*v over the (magnitude of u times the magnitude of v). We also know that θ = . Filling in that formula requires us to find the dot product of u and v, which is

u·v = (2×(-6)) + (a×8) so

u·v = -12 + 8a. Then we need the magnitudes of both u and v:

and

Putting all of together looks like this:

. Cross multiply to get

which is tricky to simplify. You really need to know the rules for multiplying radicals to do this correctly. The simplification is

and we need to solve for a. Begin by squaring both sides to get rid of the square root to get:

100(8 + 2a²) = 256a² - 768a +576 and simplify some more by distributing through the parenthesis to get

800 + 200a² = 256a² -768a + 576. Combine like terms to come up with

0 = 56a² - 768a - 224 and we need to factor that. Assuming since you're this far in math (pre-calc or maybe late algebra 2) you know how to factor, when you do, you get values for a of - .285714... and 14.

Therefore, the value for a is 14. I checked it...it works

A. 91 ft

Step-by-step explanation:

The path is 2ft wide and the pool's radius is 12.5ft (radius is half diameter, diameter is 25ft). This means that the radius of the circle created by the path is 14.5ft (2+12.5).

Knowing the radius, we can find the circumference.

Circumference = 2πr

Circumference = 2π(14.5) = 91.10618695ft ≈ 91ft