Question

If for \(\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}\), \(\vec{b} = \hat{i} - 2\hat{j} + \hat{k}\), and \(\vec{c} = -3\hat{i} + \hat{j} + 2\hat{k}\), then find \([\vec{a} \, \vec{b} \, \vec{c}]\).

• A. -15
• B. -10
• C. -30
• D. -5

Hint:

N/A

Answer 1

The Correct Option is C

The scalar triple product is given by the determinant of a matrix whose rows (or columns) are the components of vectors \(\vec{a}, \vec{b}, \vec{c}\). That is, \[ [\vec{a} \, \vec{b} \, \vec{c}] = \begin{vmatrix} 2 & 3 & 1
1 & -2 & 1
-3 & 1 & 2
\end{vmatrix} \] Step 1: Expand the determinant
Use the first row to expand the determinant:
\[ = 2 \begin{vmatrix} -2 & 1
1 & 2 \end{vmatrix} - 3 \begin{vmatrix} 1 & 1
-3 & 2 \end{vmatrix} + 1 \begin{vmatrix} 1 & -2
-3 & 1 \end{vmatrix} \] Step 2: Evaluate each minor
\[ = 2((-2)(2) - (1)(1)) = 2(-4 - 1) = 2(-5) = -10
= -3((1)(2) - (1)(-3)) = -3(2 + 3) = -3(5) = -15
= 1((1)(1) - (-2)(-3)) = 1(1 - 6) = 1(-5) = -5 \] Step 3: Add the results
\[ [\vec{a} \, \vec{b} \, \vec{c}] = -10 -15 -5 = -30 \]