If (( a × b )^2+( a · b )^2=676) and (| b |=2) then (| a |) is equal to

Correct answer: A. 13 aligned &( a × b )^2+( a · b )^2=676 &(| a | ·| b | θ n )^2+(| a | ·| b | θ)^2=676 & a² b² ^2 θ+ a^2 b² ^2 θ=676[( n )^2=1] & a² b²( ^2 θ+ ^2 θ)=676 a²=(676)/(b^2)=(676)/(4) &| a |= (676)/(4) | a |=(26)/(2) | a |=13 aligned )

BITSAT 2010

Exam: BITSAT
Year: 2010
Subject: Mathematics
Chapter: Vector Algebra

BITSAT 2010 · Mathematics · Vector Algebra

If (( a × b )^2+( a · b )^2=676) and (| b |=2) then (| a |) is...

If ((\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=676) and (|\vec{b}|=2) then (|\vec{a}|) is equal to

Options

A13
B26
C39
DNone of these

Correct answer

A. 13

Step-by-step solution

$\begin{aligned}&(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=676 \\ &(|\vec{a}| \cdot|\vec{b}| \sin \theta \hat{n})^2+(|\vec{a}| \cdot|\vec{b}| \cos \theta)^2=676 \\ & \Rightarrow a² b² \sin ^2 \theta+ a^2 b² \cos ^2 \theta=676\left[(\hat{n})^2=1\right] \\ & a² b²\left(\sin ^2 \theta+\cos ^2 \theta\right)=676 \Rightarrow a²=\frac{676}{b^2}=\frac{676}{4}\\ &|\vec{a}|=\sqrt{\frac{676}{4}} \Rightarrow|\vec{a}|=\frac{26}{2} \Rightarrow|\vec{a}|=13\end{aligned}

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