Let A = (0, 1, 5) ,B = (-sqr14,5,1) . Find the angle between A and B.

Answer 1

To find the angle between two vectors, you can use the dot product formula:

A · B = |A| |B| cos(θ)

where A · B is the dot product of A and B, |A| and |B| are the magnitudes of A and B, and θ is the angle between them.

First, let's calculate the magnitudes of A and B:

|A| = sqrt(0^2 + 1^2 + 5^2) = sqrt(26)

|B| = sqrt((-sqrt(14))^2 + 5^2 + 1^2) = sqrt(14 + 25 + 1) = sqrt(40) = 2sqrt(10)

Now, let's calculate the dot product of A and B:

A · B = (0)(-sqrt(14)) + (1)(5) + (5)(1) = 0 - sqrt(14) + 5 = 5 - sqrt(14)

Now, substitute these values into the formula and solve for θ:

5 - sqrt(14) = sqrt(26) * 2sqrt(10) * cos(θ)

cos(θ) = (5 - sqrt(14)) / (2 sqrt(26) sqrt(10))

Now, use a calculator to find the value of cos(θ):

cos(θ) ≈ -0.0428

To find the angle θ, use the inverse cosine function:

θ ≈ cos^(-1)(-0.0428)

θ ≈ 91.02 degrees

Therefore, the angle between vector A and vector B is approximately 91.02 degrees.