If \overset{\rightarrow}{a} \textrm{ } and \textrm{ } \overset{\rightarrow}{b} are vectors such that \left|\right. \textrm{ } \overset{\rightarrow}{a} + \overset{\rightarrow}{b} \textrm{ } \left|\right. = \sqrt{29} and \overset{\rightarrow}{a} \times \left(\right. 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \left.\right) = \left(\right. 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \left.\right) \times \overset{\rightarrow}{b}, then a possible value of \left(\right. \textrm{ } \overset{\rightarrow}{a} + \overset{\rightarrow}{b} \textrm{ } \left.\right) \textrm{ } \cdot \textrm{ } \left(\right. \textrm{ } - 7 \hat{i} + 2 \hat{j} + 3 \hat{k} \textrm{ } \left.\right) is

Engineering

Mathematics

Vector Product of Two Vectors

Question

If $\vec{a} and \vec{b}$ are vectors such that $| \vec{a} + \vec{b} | = \sqrt{29}$ and $\vec{a} \times (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \times \vec{b}$ , then a possible value of $(\vec{a} + \vec{b}) \cdot (- 7 \hat{i} + 2 \hat{j} + 3 \hat{k})$ is

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$Let (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) be \vec{r}$

$Hence, = \vec{a} \times \vec{r} = \vec{r} \times \vec{b} \Rightarrow (\vec{a} + \vec{b}) \times \vec{r} = 0 \Rightarrow (\vec{a} + \vec{b}) = λ \vec{r}$

$λ = \pm 1 ∵ | \vec{a} + \vec{b} | = \sqrt{29}$

$Hence, (\vec{a} + \vec{b}) = \pm (2 \hat{i} + 3 \hat{j} + 4 \hat{k})$

$\Rightarrow (\vec{a} + \vec{b}) \cdot (- 7 \hat{i} + 2 \hat{j} + 3 \hat{k}) = \pm 4$