The angular separation between the vectors A = 4i + 3j + 5k and B = i – 2j + 2k is (in degrees)

To find the angular separation between the two vectors ( A = 4i + 3j + 5k ) and ( B = i – 2j + 2k ), we can use the formula for the cosine of the angle ( theta ) between two vectors:

[

cos(theta) = frac{A cdot B}{|A| |B|}

]

1. Calculate the dot product ( A cdot B ):

[

A cdot B = (4)(1) + (3)(-2) + (5)(2) = 4 – 6 + 10 = 8

]

2. Calculate the magnitudes of ( A ) and ( B ):

[

|A| = sqrt{4^2 + 3^2 + 5^2} = sqrt{16 + 9 + 25} = sqrt{50} = 5sqrt{2}

]

[

|B| = sqrt{1^2 + (-2)^2 + 2^2} = sqrt{1 + 4 + 4} = sqrt{9} = 3

]

3. Substituting into the cosine formula:

[

cos(theta) = frac{8}{(5sqrt{2})(3)}