Question

If $\hat{a}, \hat{b}, \hat{c}$ are three mutually perpendicular unit vectors, then $\hat{a} \cdot (\hat{b} \times \hat{c})$ can take the value(s):

• A. 0
• B. 1
• C. -1
• D. $\infty$

Hint:

The scalar triple product of three mutually perpendicular unit vectors is always $\pm 1$. Its sign depends on the handedness of the coordinate system.

Answer 1

The Correct Option is B, C

Step 1: Understanding the geometry.
The vectors $\hat{a}, \hat{b}, \hat{c}$ are mutually perpendicular unit vectors. This means they behave like the standard basis vectors: \[ |\hat{a}| = |\hat{b}| = |\hat{c}| = 1, \qquad \hat{a} \perp \hat{b}, \hat{b} \perp \hat{c}, \hat{c} \perp \hat{a}. \] Step 2: Interpret the expression.
The term \[ \hat{a} \cdot (\hat{b} \times \hat{c}) \] is a scalar triple product. It represents the signed volume of the parallelepiped formed by the three vectors. Step 3: Triple product for orthonormal triads.
For any three perpendicular unit vectors, \[ \hat{b} \times \hat{c} \] is a unit vector perpendicular to both, but its direction depends on whether the set $(\hat{a},\hat{b},\hat{c})$ forms a right-handed or left-handed system.
- If right-handed → triple product = +1
- If left-handed → triple product = –1
Step 4: Conclusion.
Since both right-handed and left-handed orthonormal systems are possible, the expression can take values 1 and -1 only. Therefore the correct answers are (B) and (C).