Question

For any two vectors a and b, prove that

(a X b)2 = a2b2 (a.b)2

Answer 1

a x b= a b sin∅ .n, where a=lal and b=lbl
(а х b)² = a² b² sin² (n)² = a² b² sin²∅
a²b²(1-cos²∅)=a² b² - a² b² cos²∅
lal² lbl² -(a.b)²
(∵ (n)²=1)

Answer 2

Proof that $(a\times b)^2 = a^2b^2 \ (a.b)^2$

Proof:

For two vectors a and b,

The cross product of a and b is,

$a \times b=|a||b| \sin \theta$ ………… (i)

The dot product of a and b is,

$a . b=|a||b| \cos \theta$ ……………….. (ii)

Now,  

$\begin{aligned}(a \times b)^{2} &=(|a||b| \sin \theta)^{2} \\\ &=a^{2} \times b^{2} \times \sin ^{2} \theta \\\ &=a^{2} \times b^{2} \times\left(1-\cos ^{2} \theta\right) \\\ &=a^{2} \times b^{2}-a^{2} \times b^{2} \times\left(\cos ^{2} \theta\right) \end{aligned}$

$\Rightarrow a^{2} b^{2}-(a \cdot b)^{2}$ = R.H.S (//from equation (ii))

Hence Proved.