Use the given vectors to find the indicated expression.
-$$\mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { w } = - 4 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { u } = 5 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k }$$
Find $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$.
A) 48
B) 24
C) 58
D) $- 24$

Question

Quizplus · Accepted Answer

The expression $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ is a scalar triple product, which gives the volume of the parallelepiped formed by the vectors $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$. To find $\mathbf { v } \times \mathbf { w }$, we calculate the cross product of $\mathbf{v}$ and $\mathbf{w}$, and then dot it with $\mathbf{u}$.$\mathbf { v } \times \mathbf { w } = \begin{vmatrix} \mathbf { i } & \mathbf { j } & \mathbf { k } \ 2 & 2 & 2 \ -4 & -3 & 4 \end{vmatrix} = (2\cdot4 - 2\cdot(-3))\mathbf{i} - (2\cdot4 - 2\cdot(-4))\mathbf{j} + (2\cdot(-3) - 2\cdot(-4))\mathbf{k} = (8 + 6)\mathbf{i} - (8 + 8)\mathbf{j} + (-6 + 8)\mathbf{k} = 14\mathbf{i} - 16\mathbf{j} + 2\mathbf{k}$Then, $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } ) = (5\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}) \cdot (14\mathbf{i} - 16\mathbf{j} + 2\mathbf{k}) = 5\cdot14 + 2\cdot(-16) + 5\cdot2 = 70 - 32 + 10 = 48$.Therefore, the correct answer is 48.

Use the Given Vectors to Find the Indicated Expression Find u⋅(v×w)\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )u⋅(v×w)

Use the Given Vectors to Find the Indicated Expression Find $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$