a‾,b‾ are vectors such that |a‾+b‾|=29 and a‾×(2i‾+3j‾+4k‾)=(2i‾+3j‾+4k‾)×b‾ then possible value of (a‾+b‾)⋅(-7i‾+2j‾+3k‾)=

Let x‾=2i‾+3j‾+4K‾a‾×(2i‾+3j‾+4k‾)=(2i‾+3j‾+4k‾)×b‾ ⇒a‾×x=x×b‾ ⇒(a‾+b‾)×x‾=0 ⇒x‾=λ(a‾+b‾)|x‾|=λ|a‾+b‾| ⇒29=λ29⇒λ=1 ⇒x= a‾+b‾=2i‾+3j‾+4k‾ now (a‾+b‾)⋅(-7i‾+2j‾+3k‾)⇒-14+6+12=4

a‾,b‾ are vectors such that |a‾+b‾|=29 and a‾×(2i‾+3j‾+4k‾)=(2i‾+3j‾+4k‾)×b‾ then possible value of (a‾+b‾)⋅(-7i‾+2j‾+3k‾)=

Let x‾=2i‾+3j‾+4K‾a‾×(2i‾+3j‾+4k‾)=(2i‾+3j‾+4k‾)×b‾ ⇒a‾×x=x×b‾ ⇒(a‾+b‾)×x‾=0 ⇒x‾=λ(a‾+b‾)|x‾|=λ|a‾+b‾| ⇒29=λ29⇒λ=1 ⇒x= a‾+b‾=2i‾+3j‾+4k‾ now (a‾+b‾)⋅(-7i‾+2j‾+3k‾)⇒-14+6+12=4

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

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Detailed Solution

Let $\overline{x} = 2 \overline{i} + 3 \overline{j} + 4 \overline{K}$
$\overline{a} \times (2 \overline{i} + 3 \overline{j} + 4 \overline{k}) = (2 \overline{i} + 3 \overline{j} + 4 \overline{k}) \times \overline{b} \Rightarrow \overline{a} \times x = x \times \overline{b} \Rightarrow (\overline{a} + \overline{b}) \times \overline{x} = 0 \Rightarrow \overline{x} = λ (\overline{a} + \overline{b})$

$\begin{array}{l} | \overline{x} | = λ | \overline{a} + \overline{b} | \end{array} \Rightarrow \sqrt{29} = λ \sqrt{29} \Rightarrow λ = 1 \Rightarrow x = \overline{a} + \overline{b} = 2 \overline{i} + 3 \overline{j} + 4 \overline{k}$

$\begin{array}{ll} n o w & (\overline{a} + \overline{b}) \cdot (- 7 \overline{i} + 2 \overline{j} + 3 \overline{k}) \\ \Rightarrow & - 14 + 6 + 12 = 4 \end{array}$