Given points A (1; -2; 3) B (2; 0; -4) C (5; 7; -3) D (0; 2; 0) calculate the dot product

| September 3, 2021 | education

Given points A (1; -2; 3) B (2; 0; -4) C (5; 7; -3) D (0; 2; 0) calculate the dot product (2AB vector + CD vector) * BD vector …

The task gives four points A (1; -2; 3), B (2; 0; -4), C (5; 7; -3) and D (0; 2; 0) in three-dimensional space. It is required to calculate the scalar product of vectors 2 * AB + CD and BD, which we denote by P.
First of all, using the coordinates of these points A, B, C and D, we calculate the coordinates of the vectors participating in the scalar product P = (2 * AB + CD) * BD. We have: AB = {2 – 1; 0 – (-2); -4 – 3} = {1; 2; -7}, CD = {0 – 5; 2 – 7; 0 – (-3)} = {-5; -5; 3} and BD = {0 – 2; twenty; 0 – (-4)} = {-2; 2; 4}. First, let’s calculate the coordinates of the vector 2 * AB according to the rules of multiplying a scalar by a vector: 2 * AB = 2 * {1; 2; -7} = {2 * 1; 2 * 2; 2 * (-7)} = {2; 4; -fourteen}.
Now, according to the rules of vector addition, add the vector CD to the resulting vector 2 * AB. We have: 2 * AB + CD = {2; 4; -14} + {-5; -5; 3} = {2 + (-5); 4 + (-5); -14 + 3} = {-3; -1; -eleven}. Finally, according to the rules of scalar multiplication of vectors, we calculate the required scalar product P = (2 * AB + CD) * BD = {-3; -1; -11} * {-2; 2; 4} = (-3) * (-2) + (-1) * 2 + (-11) * 4 = 6 – 2 – 44 = -40.
Answer: -40.



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