Coordinates of points A (3; 2; 1), B (-2; 0; 1) Find the angle between the vector AB and CD, if C (2; 4; 3), D (-3; 0; 2).

| September 6, 2021 | education

The task gives four points A (3; 2; 1), B (-2; 0; 1), C (2; 4; 3) and D (-3; 0; 2) in three-dimensional space. It is required to find the angle between the vector AB and CD. First of all, using the coordinates of these points, we determine the coordinates of the vectors AB and CD. We have: AB = {-2 – 3; 0 – 2; 1 – 1} = {-5; -2; 0} and CD = {-3 – 2; 0 – 4; 2 – 3} = {-5; -4; -1}.
Find the dot product of vectors AB and CD. We have: AB * CD = (-5) * (-5) + (-2) * (-4) + 0 * (-1) = 25 + 8 + 0 = 33. Find the lengths of the vectors: | AB | = √ ((- 5) ² + (-2) ² + 0²) = √ (25 + 4 + 0) = √ (29) and | CD | = √ ((- 5) ² + (-4) ² + (-1) ²) = √ (25 + 16 + 1) = √ (42).
Find the cosine of the angle (which we denote by α) between the vectors: cosα = (AB * CD) / (| AB | * | CD |) = 33 / (√ (29) * √ (42)) = (11√ (1218) ) / 406. Then, α = arcos ((11√ (1218)) / 406).
Answer: arcos ((11√ (1218)) / 406).



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