First slide

Dot product or scalar product of vectors

Question

Given three vectors $\vec{a}, \vec{b} and \vec{c}$ two of which are non-collinear. Further if $(\vec{a} + \vec{b})$ is collinear with $\vec{c}, (\vec{b} + \vec{c})$ is collinear with, $\vec{a}, | \vec{a} | = | \vec{b} | = | \vec{c} | = \sqrt{2} .$ Find the value of $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$

Moderate

A

3
B

-3
C

0
D

cannot be evaluated

Solution

$\begin{aligned} \vec{a} + \vec{b} = λ \vec{c} - - - (i) \\ and \vec{b} + \vec{c} = μ \vec{a} - - - (ii) \end{aligned}$
$∴ (λ \vec{c} - \vec{a}) + \vec{c} = μ \vec{a} (putting \vec{b} = λ \vec{c} - \vec{a})$
$\begin{array}{ll} \Rightarrow (λ + 1) \vec{c} = (μ + 1) \vec{a} \\ \Rightarrow λ = μ = - 1 \\ \Rightarrow \vec{a} + \vec{b} + \vec{c} = 0 \\ or | \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{c} |^{2} + 2 (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \\ or \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = - 3 \end{array}$

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