Question

Find the angle θ between the vectors in radians and in degrees. (Round your answer for part (a) to three decimal places and part (b) to one decimal place.)

u = 2i − 5j + k
v = i − 2j + k

Answer 1

To find the angle between two vectors, calculate the dot product and the magnitudes of the vectors. Then use the formula acos(u · v / (|u| * |v|)) to find the angle in radians and degrees.

Step-by-step explanation:

To find the angle between two vectors, you can use the dot product formula. For vectors u = 2i - 5j + k and v = i - 2j + k, the dot product is calculated as:

u · v = (2)(1) + (-5)(-2) + (1)(1) = 2 + 10 + 1 = 13

The magnitudes of the vectors are:

|u| = sqrt((2)^2 + (-5)^2 + 1^2) = sqrt(4 + 25 + 1) = sqrt(30) ≈ 5.477

|v| = sqrt((1)^2 + (-2)^2 + 1^2) = sqrt(1 + 4 + 1) = sqrt(6) ≈ 2.449

Using the dot product and the magnitudes, the angle between the two vectors can be calculated as:

θ = acos(u · v / (|u| * |v|)) = acos(13 / (5.477 * 2.449)) ≈ 1.872 radians or 107.275 degrees