a) Let Vector a = (3, 4, 1), Vector b = (5, -2, 3) and Vector c = (0, 1, -3). Find the triple product, Vector a • (Vector b × Vector c).
To find the triple product, Vector a • (Vector b × Vector c), we need to perform the following steps:
1. Calculate the cross product of Vector b and Vector c: (Vector b × Vector c).
- The cross product of two vectors in three-dimensional space is given by the formula:
Vector b × Vector c = (b2c3 - b3c2, b3c1 - b1c3, b1c2 - b2c1)
- Substitute the values of Vector b and Vector c:
Vector b × Vector c = ((5 × 1) - (-2 × -3), (-2 × 0) - (5 × -3), (5 × -3) - (-2 × 0))
= (11, 15, -15)
2. Take the dot product of Vector a and the result from step 1: Vector a • (Vector b × Vector c).
- The dot product of two vectors in three-dimensional space is given by the formula:
Vector a • Vector d = (a1d1 + a2d2 + a3d3)
- Substitute the values of Vector a and the result from step 1:
Vector a • (Vector b × Vector c) = (3 × 11) + (4 × 15) + (1 × -15)
= 33 + 60 - 15
= 78
Therefore, the triple product, Vector a • (Vector b × Vector c), is equal to 78.