For mutually perpendicular unit vectors a, b, c, the scalar triple product a( b c) can be

GATE 2022 AE – Question 24

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 1 mark · Multiple select

For mutually perpendicular unit vectors $\hat a,\hat b,\hat c$, the scalar triple product $\hat a\cdot(\hat b\times\hat c)$ can be

0
1
-1
$\infty$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) 1; (C) -1

Explanation

For three **mutually perpendicular unit vectors** $\hat a,\hat b,\hat c$:

$\hat b\times\hat c$ has magnitude $|\hat b||\hat c|\sin90^\circ=1$ and is perpendicular to both $\hat b$ and $\hat c$.
Since $\hat a$ is perpendicular to both $\hat b$ and $\hat c$, it is parallel (or anti-parallel) to $\hat b\times\hat c$.

So the dot product is $\hat a\cdot(\hat b\times\hat c)=\pm1$:
- **+1** if $\hat a,\hat b,\hat c$ form a right-handed set,
- **−1** if they form a left-handed set.

It is never 0 (the vectors are not coplanar) and never infinite. Answer **1 and −1** (B and C).