Question

If $ u=a - b $ and $ v = a + b $ and $ |a| = |b| = 2 $ , then $ | u \times v| $ is equal to

• A. $ 2\sqrt{16-\left(a\cdot b\right)^{2}} $
• B. $ \sqrt{16-\left(a\cdot b\right)^{2}} $
• C. $ 2\sqrt{4-\left(a\cdot b\right)^{2}} $
• D. $ 2\sqrt{4+\left(a\cdot b\right)^{2}} $

Answer 1

Answer: (A)

$| u \times v |=|( a - b ) \times( a + b )|$
$=2| a \times b |$
$(\because a \times a = b \times b =0)$
and $| a \times b |^{2}+( a \cdot b )^{2}$
$=(a b \sin \theta)^{2}+(a b \cos \theta)^{2}$
$=a^{2} b^{2}$
$\Rightarrow | a \times b |=\sqrt{a^{2} b^{2}-( a \cdot b )^{2}}$
So, $| u \times v |=2| a \times b |$
$=2 \sqrt{a^{2} b^{2}-(a \cdot b)^{2}}$
$=2 \sqrt{2^{2} 2^{2}-(a \cdot b)^{2}}$
$=2 \sqrt{16-(a \cdot b)^{2}}$
$\left(\because |a|=|b|=2\right)$