Let A = (0, 1, 5) ,B = (-sqrt 14,5,1) . Find the angle between A and B.

Answer 1

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

A · B = |A| |B| cos(θ)

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

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

|A| = √(0^2 + 1^2 + 5^2) = √26
|B| = √((-√14)^2 + 5^2 + 1^2) = √(14 + 26) = √40 = 2√10

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

A · B = (0*-sqrt14) + (1*5) + (5*1) = -√14 + 5 + 5 = 10 - √14

Now, let's substitute the values into the dot product formula and solve for θ:

10 - √14 = √26 * 2√10 * cos(θ)

Divide both sides by √26 * 2√10:

(10 - √14) / (2√26√10) = cos(θ)

Simplify the expression:

(10 - √14) / (2√(260)) = cos(θ)

Now, use a calculator to evaluate the right-hand side expression:

(10 - √14) / (2√(260)) ≈ 0.205

Next, use the inverse cosine function to find the angle θ:

θ ≈ arccos(0.205) ≈ 1.361 radians or 78.015 degrees

Therefore, the angle between vectors A and B is approximately 1.361 radians or 78.015 degrees.