First slide

Scalar triple product of vectors

Question

$A (\vec{a}), B (\vec{b}) and C (\vec{c})$ are the vertices of triangle ABC and $R (\vec{r})$ i, any point in the plane of triangle ABC, the $\vec{r} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})$ i,
is always equal to

Moderate

A

zero
B

$[\vec{a} \vec{b} \vec{c}]$
C

$- [\vec{a} \vec{b} \vec{c}]$
D

none of these

Solution

A vector perpendicular to the plane of $A (\vec{a}), B (\vec{b}) and C (\vec{c})$
$(\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$
Now for any point $R (\vec{r})$ in the plane of A, B and C is
$\begin{array}{l} \vec{(r} - \vec{a}) \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) = 0 \\ \vec{r} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) \\ - \vec{a} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) = 0 \\ \vec{r} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) \\ = \vec{0} + \vec{a} \cdot \vec{b} \times \vec{c} + \vec{0} \\ \vec{r} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) = [\vec{a} \vec{b} \vec{c}] \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App