Vector Algebra
Scalar Triple Product and Volume
Grade 12
Question:
<p>If the volume of parallelepiped formed by the vectors <i>a</i>, <i>b</i>, <i>c</i> as three coterminous edges is 27 cu units, then the volume of the parallelepiped with \(\alpha = a + 2b - c\), \(\beta = a - b\) and \(\gamma = a - b - c\) as three coterminous edges is</p>
<p>(a) 27</p>
<p>(b) 9</p>
<p>(c) 81</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The volume of a parallelepiped changes by the absolute value of the determinant of the transformation matrix when vectors are expressed as linear combinations.
Given: [ a b c ] = 27 We need to find: [α β γ] where \(\alpha = a + 2b - c\), \(\beta = a - b\), \(\gamma = a - b - c\) Using properties of scalar triple product: The new volume equals the absolute value of the determinant formed by the coefficients of the linear combination. The determinant is: \(\begin{vmatrix} 1 & 2 & -1 \\ 1 & -1 & 0 \\ 1 & -1 & -1 \end{vmatrix} = 1(1-0) - 2(-1-0) + (-1)(-1+1) = 1 + 2 + 0 = 3/9\) Therefore, volume = \(|\frac{1}{3}| \times 27 = 9\) cu units ∴ Answer is (b) 9
Correct Answer: B