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NDA Math Vector Algebra

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For the next four (04) items that follow:

\vec{a}+\vec{b}+\vec{c}=\vec{0}

and

|\vec{a}|=3,|\vec{b}|=5

and

|\vec{c}|=7

[NDA II 2014]

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Question : 96 of 101

Marks: +1, -0

What is cosine of the angle between

\overset{\rightarrow}{b}

\overset{\rightarrow}{c}

11/12
13/14
-11/12
-13/14

Solution:

\overset{\rightarrow}{b}+\overset{\rightarrow}{c}=-\overset{\rightarrow}{a}

\Rightarrow\left(\overset{\rightarrow}{b}+\overset{\rightarrow}{c}\right).\left(\overset{\rightarrow}{b}+\overset{\rightarrow}{c}\right)=-\overset{\rightarrow}{a}.-\overset{\rightarrow}{a}

\Rightarrow\overset{\rightarrow}{b}.\overset{\rightarrow}{b}+\overset{\rightarrow}{c}.\overset{\rightarrow}{c}+\overset{\rightarrow}{2b}.\overset{\rightarrow}{c}

=\overset{\rightarrow}{a}.\overset{\rightarrow}{a}

\Rightarrow2\left|\overset{\rightarrow}{b}\right|\left|\overset{\rightarrow}{c}\right|\cos\alpha=9-25-49=-65

\Rightarrow2\times5\times7\cos\alpha= -65

\Rightarrow\cos\alpha=-\frac{13}{14}

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