Question

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

Answer 1

Explanation:

The formula to find the angle between 2 vectors is

`cos\theta=(u*v)/(|u||v|)` which is the dot product of u*v over the (magnitude of u times the magnitude of v). We also know that θ =
`(√(2) )/(2)`. 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:

`|u|=√(2^2+a^2)=√(4+a^2)` and

`|v|=√((-6)^2+8^2)=√(100)=10`

Putting all of together looks like this:

`(√(2) )/(2)=(-12+8a)/(10√(4+a^2) )` . Cross multiply to get

`10√(2)(√(4+a^2))=2(-12+8a)` which is tricky to simplify. You really need to know the rules for multiplying radicals to do this correctly. The simplification is

`10√(8+2a^2)=-24+16a` 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

Answer 2

a = 14, U = (2, 14)

Explanation:

the other answer is correct.

just to add the solution path for the quadratic equation

0 = 56a² - 768a - 224

in general, a quadratic equation usually expressed in x

0 = ax² + bx + c

has its solution as

x = (-b ± sqrt(b² - 4ac))/(2a)

for us here this looks like this then

a = (768 ± sqrt(768² - 4×56×-224))/(2×56) =

= (768 ± sqrt(589824 + 4×56×224))/112 =

= (768 ± sqrt(589824 + 50176))/112 =

= (768 ± sqrt(640000))/112 = (768 ± 800)/112

a1 = (768 + 800)/112 = 1568/112 = 14

a2 = (768 - 800)/112 = -32/112 = -0.29

and the negative solution would be for a vector solution in the wrong direction (down instead of up). as V is pointing up left, and the angle is only 45 degrees (pi/4), also U has to point up.

so, 14 is the solution for U.