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Maths-

General

Easy

Question

Let $a comma b$ and $c$ be non-zero vectors such that $left parenthesis a cross times b right parenthesis cross times c equals fraction numerator 1 over denominator 3 end fraction vertical line b vertical line vertical line c vertical line a.$ If q is the acute angle between the vectors $b$ and $c$ , then $sin invisible function application theta$ equals

$\frac{2 \sqrt{2}}{3}$
$\frac{\sqrt{2}}{3}$
$\frac{2}{3}$
$\frac{1}{3}$

The correct answer is: $fraction numerator 2 square root of 2 over denominator 3 end fraction$

$left parenthesis a cross times b right parenthesis cross times c equals fraction numerator 1 over denominator 3 end fraction vertical line b vertical line vertical line c vertical line a$
Þ $left parenthesis a. c right parenthesis b minus left parenthesis b. c right parenthesis a equals fraction numerator 1 over denominator 3 end fraction vertical line b vertical line vertical line c vertical line a$
Þ $left parenthesis a. c right parenthesis b equals open curly brackets left parenthesis b. c right parenthesis plus fraction numerator 1 over denominator 3 end fraction vertical line b vertical line vertical line c vertical line close curly brackets a$
Þ $left parenthesis a. c right parenthesis b equals vertical line b vertical line vertical line c vertical line open curly brackets cos invisible function application theta plus fraction numerator 1 over denominator 3 end fraction close curly brackets a$
As $a$ and $b$ are not parallel, $a. c equals 0$ and $cos invisible function application theta plus fraction numerator 1 over denominator 3 end fraction$ =0
Þ $cos invisible function application theta equals negative fraction numerator 1 over denominator 3 end fraction$ . Hence $sin invisible function application theta equals fraction numerator 2 square root of 2 over denominator 3 end fraction$ .

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