Question

if a = 3i - j + 4k.b = 2i + 3j - k and c = 5i + 2j + 3k than ā.(ā×b)
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Answer 1

$\bar{a}\cdot (\bar{b}\times \bar{c})$ = 110

Step-by-step explanation:

The correct question is:

If $\bar{a}=3 \hat{i}-\hat{j}+4 \hat{k}, \bar{b}=2 \hat{i}+3 \hat{j}-\hat{k}, \bar{c}=-5 \hat{i}+2 \hat{j}+3 \hat{k}$, then $\bar{a} \cdot(\bar{b} \times \bar{c})$ = ____.

Given data:

$\bar{a}=3 \hat{i}-\hat{j}+4 \hat{k}$

$\bar{b}=2 \hat{i}+3 \hat{j}-\hat{k}$

$\bar{c}=-5 \hat{i}+2 \hat{j}+3 \hat{k}$

To find $\bar{a} \cdot(\bar{b} \times \bar{c})$:

Put a, b and c values in 1st, 2nd and 3rd rows of matrix determinant.

$\bar{a} \cdot(\bar{b} \times \bar{c})=\left|\begin{array}{ccc}3 & -1 & 4 \\2 & 3 & -1 \\-5 & 2 & 3\end{array}\right|$

              $=3\left|\begin{array}{cc}3 & -1 \\2 & 3 \end{array}\right| -(-1) \left|\begin{array}{cc}2 & -1 \\-5 & 3 \end{array}\right| + 4 \left|\begin{array}{cc}2 & 3 \\-5 & 2 \end{array}\right|$

Matrix formula:

$\left|\begin{array}{cc}a & b \\c & d \end{array}\right| = ad-bc$

            $=3(3 \cdot 3-(-1) \cdot 2) -(-1)(2 \cdot 3-(-1)(-5))+4(2 \cdot 2-3(-5))$

            = 3(9 + 2) + 1(6 – 5) + 4(4 + 15)

            = 3(11) +1(1) + 4(19)

            = 33 + 1 + 76

            = 110

Hence $\bar{a}\cdot (\bar{b}\times \bar{c})$ = 110.