Question
Show that $\left( {\vec a - \vec b} \right) \times \left( {\vec a + \vec b} \right) = 2\left( {\vec a \times \vec b} \right)$

Answer

L.H.S $= \left( {\vec a - \vec b} \right) \times \left( {\vec a + \vec b} \right)$

$= \vec a \times \vec a + \vec a \times \vec b - \vec b \times \vec a - \vec b \times \vec b$

$= 0 + \vec a \times \vec b + \vec a \times \vec b - 0$    $[as \ \vec a×\vec b=-\vec b×\ \vec a]$

$=2(\vec a×\vec b)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Similar questions

Write the position vector of a point dividing the line segment joining points A and B with position vectors $\vec{\text{a}}\text{ and }\vec{\text{b}}$ externally in the ratio 1 : 4, where $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}$ and $\vec{\text{b}}=-\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$.
Evaluate the following integrals:
$\int\limits^{-\frac{\pi}{2}}_{-\frac{3\pi}{2}}\Big\{\sin^2(3\pi+\text{x})+(\pi+\text{x})^3\Big\}\text{dx}$
Evaluate the following definite integrals:
$\int_{-2}^\limits{3}\frac{1}{\text{x}+7} \text{ dx}$
If $\begin{bmatrix}\text{a+b}&2\\5&\text{b} \end{bmatrix}=\begin{bmatrix}6&5\\2&2 \end{bmatrix}$, then find a.
If $\begin{bmatrix}2\text{x+y}&3\text{y}\\0&4 \end{bmatrix}=\begin{bmatrix}6&0\\6&4\end{bmatrix}$, then find x.
If A is a skew-symmetric and n ∈ N such that $(\text{A}^\text{n})^\text{T}=\lambda\text{A}^\text{n},$ write the value of $\lambda.$
Find the angle between the vectors $\vec{\text{a}} $ and $\vec{\text{b}},$ where

$\vec{\text{a}}=2\hat {\text{i}}-\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}} =4\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=2\text{x}^2-5\text{x}+3\text{ on }[1,3]$
Write the distances of the point (7, -2, 3) from XY, YZ and XZ-planes.
For the principal values, evaluate the following:
$\text{cosec}^{-1}\Big(2\tan\frac{11\pi}{6}\Big)$