Question

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{p}, \vec{q}, \vec{r}$ are vectors defined by:}\\ \[ \vec{p} = \frac{\vec{a} \times \vec{c}}{[\vec{a} \, \vec{b} \, \vec{c}]}, \quad \vec{q} = \frac{\vec{c} \times \vec{b}}{[\vec{a} \, \vec{b} \, \vec{c}]}, \quad \vec{r} = \frac{\vec{b} \times \vec{a}}{[\vec{a} \, \vec{b} \, \vec{c}]}, \] then $(\vec{i} + \vec{b}) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r}$ is:

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is D

The scalar triple product $[\vec{a} \, \vec{b} \, \vec{c}] = 1$ (normalization). Each term evaluates to 1 because the dot product of each vector pair is 1. Summing the terms gives: \[ 1 + 1 + 1 = 3. \]