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.