Question

Let
A=i−2j−3k
,
B=2i+3j+k
and
C=i+3j−2k
, compute
A.(B×C)
15
5
10
20

Answer 1

Answer on Question #56114 – Math – Vector Calculus

1. Let $A = i - 2j - 3k$, $B = 2i + 3j + k$ and $C = i + 3j - 2k$; compute $A \cdot (B \times C)$.

Solution

Vector (cross) product can be rewritten in matrix form and computed:

$$\begin{array}{l} \boldsymbol{B} \times \boldsymbol{C} = \left| \begin{array}{ccc} \boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\ 2 & 3 & 1 \\ 1 & 3 & -2 \end{array} \right| = \boldsymbol{i} \left| \begin{array}{cc} 3 & 1 \\ 3 & -2 \end{array} \right| - \boldsymbol{j} \left| \begin{array}{cc} 2 & 1 \\ 1 & -2 \end{array} \right| + \boldsymbol{k} \left| \begin{array}{cc} 2 & 3 \\ 1 & 3 \end{array} \right| = (-6 - 3)\boldsymbol{i} + (4 + 1)\boldsymbol{j} + \\ + (6 - 3)\boldsymbol{k} = -9\boldsymbol{i} + 5\boldsymbol{j} + 3\boldsymbol{k}. \end{array}$$

Now we can compute:

$$\begin{array}{l} \boldsymbol{A} \cdot (\boldsymbol{B} \times \boldsymbol{C}) = (\boldsymbol{i} - 2\boldsymbol{j} - 3\boldsymbol{k}) \cdot (-9\boldsymbol{i} + 5\boldsymbol{j} + 3\boldsymbol{k}) = 1 \cdot (-9) - 2 \cdot 5 - 3 \cdot 3 = -9 - 10 - \\ -9 = -28. \end{array}$$

Answer: $A \cdot (B \times C) = -28$.

www.AssignmentExpert.com