If a, b and c are $$non-zero$$ vectors. The value of a⋅$${(b+c)$$×$$(a+b+c)}$$ is $$_____________$$

a⋅$${(b+c)$$×(a+b+c)}$$ or a⋅$${(b+c)×$$a+(b+c)$$×(b+c)}$$ or a⋅$${(b+c)×$$a}$$ But (b+c)×a is perpendicular to a. ∴a⋅$${(b+c)$$×$$a}=0$$

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If a, b and c are non-zero vectors. The value of $a \cdot {(b + c) \times (a + b + c)}$ is _____________

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Solution

$\begin{array}{l} \begin{aligned} a \cdot {(b + c) \times (a + b + c)} \\ or a \cdot {(b + c) \times a + (b + c) \times (b + c)} \end{aligned} \\ or a \cdot {(b + c) \times a} \\ But (b + c) \times a is perpendicular to a . \\ ∴ a \cdot {(b + c) \times a} = 0 \end{array}$

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