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Maths-

General

Easy

Question

Let $stack a with rightwards arrow on top equals 2 stack i with hat on top plus stack j with hat on top minus 2 stack k with hat on top$ and $stack b with rightwards arrow on top equals stack i with hat on top plus stack j with hat on top$ . If $stack c with rightwards arrow on top$ is a vector such that $stack a with rightwards arrow on top. stack c with rightwards arrow on top equals open vertical bar stack c with rightwards arrow on top close vertical bar comma open vertical bar stack c with rightwards arrow on top minus stack a with rightwards arrow on top close vertical bar$ = $2 square root of 2$ and the angle between $open parentheses stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close parentheses$ and $stack c with rightwards arrow on top$ is 30⁰, then find the value of $open vertical bar open parentheses stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close parentheses cross times stack c with rightwards arrow on top close vertical bar$ .

$\frac{2}{3}$
$\frac{3}{2}$
2
3

The correct answer is: $3 over 2$

$open vertical bar open parentheses stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close parentheses cross times stack c with rightwards arrow on top close vertical bar$ = $open vertical bar stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close vertical bar short rightwards arrow over leftwards arrow vertical line stack c with rightwards arrow on top vertical line$ sin 30⁰
$stack a with rightwards arrow on top cross times stack b with rightwards arrow on top equals open vertical bar table row i j k row 2 1 cell negative 2 end cell row 1 1 0 end table close vertical bar equals negative 2 i plus 2 j plus k$ Þ $vertical line stack a with rightwards arrow on top cross times stack b with rightwards arrow on top vertical line equals 3$
$vertical line stack c with ⃑ on top minus stack a with ⃑ on top vertical line to the power of 2 end exponent$ = 8 Þ $vertical line stack c with ⃑ on top vertical line to the power of 2 end exponent plus vertical line stack a with ⃑ on top vertical line to the power of 2 end exponent minus 2 stack c with ⃑ on top. stack a with ⃑ on top$ = 8
Þ $vertical line stack c with ⃑ on top vertical line to the power of 2 end exponent minus 2 vertical line stack c with ⃑ on top vertical line plus 1 equals 0$ Þ $vertical line stack c with ⃑ on top vertical line equals 1$ Þ $open vertical bar open parentheses stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close parentheses cross times stack c with rightwards arrow on top close vertical bar$ = 1. 3. $fraction numerator 1 over denominator 2 end fraction$ = $fraction numerator 3 over denominator 2 end fraction$ .

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