Vector Algebra
Scalar Triple Product and Tetrahedron
Grade 12
Question:
<p>If <math>a</math>, <math>b</math> and <math>c</math> are three non-coplanar uni-modular vectors, each inclined with other at an angle <math>30°</math>, then volume of tetrahedron whose edges are <math>a</math>, <math>b</math> and <math>c</math> is</p>
<p>(a) <math>\frac{3\sqrt{3} - 5}{12}</math></p>
<p>(b) <math>\frac{3\sqrt{3} - 5}{12}</math></p>
<p>(c) <math>\frac{5\sqrt{2} + 3}{12}</math></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The volume of a tetrahedron with edges a, b, c is (1/6)|[abc]|, which can be computed using the scalar triple product in terms of dot products using the Gram determinant.
Solution: Since the volume of tetrahedron with edges <math>a</math>, <math>b</math> and <math>c</math> is <math>V = \frac{1}{6}[abc]</math> Given: <math>a \cdot a = b \cdot b = c \cdot c = 1</math> (uni-modular) and <math>a \cdot b = b \cdot c = c \cdot a = \cos(30°) = \frac{\sqrt{3}}{2}</math> <math>V^2 = \frac{1}{36}[abc]^2 = \frac{1}{36}\begin{vmatrix} a \cdot a & a \cdot b & a \cdot c \\ b \cdot a & b \cdot b & b \cdot c \\ c \cdot a & c \cdot b & c \cdot c \end{vmatrix}</math> <math>= \frac{1}{36}\begin{vmatrix} 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & 1 & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 1 \end{vmatrix}</math> Evaluating the determinant gives <math>V = \frac{3\sqrt{3} - 5}{12}</math>
Correct Answer: A