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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
If $\mathbf{a},\; \mathbf{b},\; \mathbf{c}$ are unit vectors such that
$\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$,
then the value of
$\mathbf{a}\cdot \mathbf{b} + \mathbf{b}\cdot \mathbf{c} + \mathbf{c}\cdot \mathbf{a}$ is:
Solution
Given: $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$ Square both sides:
$(\mathbf{a} + \mathbf{b} + \mathbf{c})\cdot(\mathbf{a} + \mathbf{b} + \mathbf{c}) = 0$
$\mathbf{a}\cdot\mathbf{a} + \mathbf{b}\cdot\mathbf{b} + \mathbf{c}\cdot\mathbf{c} + 2(\mathbf{a}\cdot\mathbf{b} + \mathbf{b}\cdot\mathbf{c} + \mathbf{c}\cdot\mathbf{a}) = 0$
Since each of $\mathbf{a},\;\mathbf{b},\;\mathbf{c}$ is a unit vector:
$\mathbf{a}\cdot\mathbf{a} = \mathbf{b}\cdot\mathbf{b} = \mathbf{c}\cdot\mathbf{c} = 1$
Thus:
$3 + 2(\mathbf{a}\cdot\mathbf{b} + \mathbf{b}\cdot\mathbf{c} + \mathbf{c}\cdot\mathbf{a}) = 0$
$2(\mathbf{a}\cdot\mathbf{b} + \mathbf{b}\cdot\mathbf{c} + \mathbf{c}\cdot\mathbf{a}) = -3$
$\mathbf{a}\cdot\mathbf{b} + \mathbf{b}\cdot\mathbf{c} + \mathbf{c}\cdot\mathbf{a} = -\dfrac{3}{2}$
Therefore:
${-\dfrac{3}{2}}$
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