If \overset{\rightarrow}{a} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{b} are vectors in space given by \overset{\rightarrow}{a} \textrm{ } = \frac{\hat{i} - 2 \hat{j}}{\sqrt{5}} \textrm{ }\textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{b} \textrm{ } = \textrm{ } \frac{2 \hat{i} + \hat{j} + 3 \hat{k}}{\sqrt{14}} , then the value of \left(\right. 2 \overset{\rightarrow}{a} + \overset{\rightarrow}{b} \left.\right) . \textrm{ } \left(\left[\right. \left(\right. \overset{\rightarrow}{a} \times \overset{\rightarrow}{b} \left.\right) - 2 \left(\right. \overset{\rightarrow}{a} \times \overset{\rightarrow}{b} \left.\right) \times \overset{\rightarrow}{b} \left]\right.\right)

Engineering

Mathematics

Scalar Triple Product

Question

If $\vec{a} and \vec{b}$ are vectors in space given by $\vec{a} = \frac{\hat{i} - 2 \hat{j}}{\sqrt{5}} and \vec{b} = \frac{2 \hat{i} + \hat{j} + 3 \hat{k}}{\sqrt{14}}$ , then the value of $(2 \vec{a} + \vec{b}) . {[(\vec{a} \times \vec{b}) - 2 (\vec{a} \times \vec{b}) \times \vec{b}]}_{}^{}$ , is

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$(2 \vec{a} + \vec{b}) {[(\vec{a} \times \vec{b}) - 2 (\vec{a} \times \vec{b}) \times \vec{b}]}_{}^{} = (2 \vec{a} + \vec{b}) {[(\vec{a} \cdot \vec{a}) \vec{b} - (\vec{b} \cdot \vec{a}) \vec{a} - 2 (\vec{a} \cdot \vec{b}) \vec{b} + 2 (\vec{b} \cdot \vec{b}) \vec{a}]}_{}^{}$

$= (2 \vec{a} + \vec{b}) {[\vec{b} - \vec{a} - 2 \vec{b} + 2 \vec{a}]}_{}^{} = (2 \vec{a} + \vec{b}) \cdot (2 \vec{a} + \vec{b}) = 4 a^{2} + b^{2} + 0 = 5$