Find the triple scalar product (u x v) · w of the given vectors.
-u = 4i + 2j - k; v = 2i + 8j - 6k; w = 7i + 4j - 5k
A) 48
B) -96
C) -216
D) -80

Question

Quizplus · Accepted Answer

The triple scalar product $(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}$ can be calculated using the determinant of a matrix formed by the components of vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ as rows. For vectors $\mathbf{u} = 4\mathbf{i} + 2\mathbf{j} - \mathbf{k}$, $\mathbf{v} = 2\mathbf{i} + 8\mathbf{j} - 6\mathbf{k}$, and $\mathbf{w} = 7\mathbf{i} + 4\mathbf{j} - 5\mathbf{k}$, the determinant is:$$\begin{vmatrix}4 & 2 & -1 \2 & 8 & -6 \7 & 4 & -5 \\end{vmatrix}$$Calculating this determinant gives:$$4(8(-5) - 4(-6)) - 2(2(-5) - 7(-6)) + (-1)(2(4) - 7(8)) = 4(-40 + 24) - 2(-10 + 42) - (8 - 56) = 4(-16) - 2(32) - (-48) = -64 - 64 + 48 = -128 + 48 = -80.$$Therefore, the triple scalar product $(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}$ is -80.

Find the Triple Scalar Product (U X V) · W