Calculate the angle between u = [5,7,-1] and v = [8,7,8]

For any two vectors u and v

u.v = |u| |v|cosØ, where Ø is the angle between them
40 + 49 - 8 = √(25+49+1)√(64+49+64)cosØ
81 = √75√177 cosØ
cos Ø = .70302
Ø = 45.33°

thank you!

To calculate the angle between two vectors, we can use the dot product formula:

u · v = ||u|| ||v|| cos(θ)

where u · v is the dot product of u and v, ||u|| and ||v|| are the magnitudes of u and v respectively, and θ is the angle between the two vectors.

First, let's find the magnitudes of u and v:

||u|| = √(5^2 + 7^2 + (-1)^2) = √(25 + 49 + 1) = √75 = 5√3
||v|| = √(8^2 + 7^2 + 8^2) = √(64 + 49 + 64) = √177 = √(9 * 19) = 3√19

Next, let's calculate the dot product of u and v:

u · v = 5 * 8 + 7 * 7 + (-1) * 8 = 40 + 49 - 8 = 81

Substituting the values into the dot product formula, we have:

81 = (5√3)(3√19) * cos(θ)

To solve for θ, let's isolate cos(θ):

cos(θ) = 81 / (5√3 * 3√19)

To simplify this expression, multiply the denominator:

cos(θ) = 81 / (15√57)

Now, let's divide both numerator and denominator by 3:

cos(θ) = 27 / (5√57)

To find the angle θ, we can take the inverse cosine (arccos) of both sides:

θ = arccos(27 / (5√57))

Using a calculator, we can evaluate this expression to find the angle θ.