Question

If \(\vec{a}, \vec{b}, \vec{c}\) are unit vectors and \(\vec{a} \perp \vec{b}\), and \((\vec{a} - \vec{c}) \cdot (\vec{b} + \vec{c}) = 0\), and \(\vec{c} = l\vec{a} + m\vec{b} + n(\vec{a} \times \vec{b})\), then \(n^2 =\)

• A. \(l^2 + m^2\)
• B. \(-\dfrac{2}{m}\)
• C. \(2l - 2m\)
• D. \(\dfrac{l}{m} + l + m\)

Hint:

Break vectors into orthogonal basis and use scalar product conditions to find unknown coefficients.

Answer 1

The Correct Option is B

From \(\vec{c} = l\vec{a} + m\vec{b} + n(\vec{a} \times \vec{b})\), compute: \[ (\vec{a} - \vec{c}) \cdot (\vec{b} + \vec{c}) = 0 \Rightarrow \text{Dot both expressions and simplify using } \vec{a} \cdot \vec{b} = 0, |\vec{a}| = |\vec{b}| = 1. \] Final result gives: \[ n^2 = -\dfrac{2}{m} \]