First slide

Scalar triple product of vectors

Question

$\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} and \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$ where $\vec{a}, \vec{b} and \vec{c}$ are thee non-coplanar vectors, then the value of the expression $(\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r})$ is

Moderate

A

3
B

2
C

1
D

0

Solution

We have,
$\begin{array}{l} \vec{a} \cdot \vec{p} = \vec{a} \cdot \frac{(\vec{b} \times \vec{c})}{[\vec{a} \vec{b} \vec{c}]} = \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{[\vec{a} \vec{b} \vec{c}]} = \frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} \vec{c}]} = 1 \\ \vec{a} \cdot \vec{q} = \vec{a} \cdot \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} = \frac{[\vec{a} \vec{c} \vec{a}]}{[\vec{a} \vec{b} \vec{c}]} = 0 \end{array}$
$Similarly, \vec{a} \cdot \vec{r} = 0, \vec{b} \cdot \vec{p} = 0, \vec{b} \cdot \vec{q} = 1, \vec{b} \cdot \vec{r} = 0$
$\begin{aligned} \vec{c} \cdot \vec{p} = 0, \vec{c} \cdot \vec{q} = 0 and \vec{c} \cdot \vec{r} = 1 \\ ∴ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r}) \end{aligned}$
$\begin{array}{l} = \vec{a} \cdot \vec{p} + \vec{a} \cdot \vec{q} + \vec{a} \cdot \vec{r} + \vec{b} \cdot \vec{p} + \vec{b} \cdot \vec{q} + \vec{b} \cdot \vec{r} + \vec{c} \cdot \vec{p} + \vec{c} \cdot \vec{q} + \vec{c} \cdot \vec{r} \\ = 1 + 1 + 1 = 3 \end{array}$

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