If a¯ and b¯ are vectors such that a¯+b¯=29 and a¯×2i¯+3j¯+4k¯=2i¯+3j¯+4k¯×b¯, then a possible value of a¯+b¯.−7i¯+2j¯+3k¯ is

|a¯+b¯|=29 , a¯×(2i¯+3j¯+4k¯)=(2i¯+3j¯+4k¯)×b¯m¯×n¯=n¯×p¯⇒m¯=p¯=kn¯ then a¯=b¯=k(2i¯+3j¯+4k¯)|k(2i¯+3j¯+4k¯)+k(2i¯+3j¯+4k¯)|=29⇒|2k(2i¯+3j¯+4k¯)|=292k29=29⇒k=12⇒a¯=b¯=12(2i¯+3j¯+4k¯)(a¯+b¯)⋅(−7i¯+2j¯+3k¯)=(2i¯+3j¯+4k¯)⋅(−7i¯+2j¯+3k¯)(a¯+b¯)⋅(−7i¯+2j¯+3k¯)=−14+6+12=4

If a¯ and b¯ are vectors such that a¯+b¯=29 and a¯×2i¯+3j¯+4k¯=2i¯+3j¯+4k¯×b¯, then a possible value of a¯+b¯.−7i¯+2j¯+3k¯ is

|a¯+b¯|=29 , a¯×(2i¯+3j¯+4k¯)=(2i¯+3j¯+4k¯)×b¯m¯×n¯=n¯×p¯⇒m¯=p¯=kn¯ then a¯=b¯=k(2i¯+3j¯+4k¯)|k(2i¯+3j¯+4k¯)+k(2i¯+3j¯+4k¯)|=29⇒|2k(2i¯+3j¯+4k¯)|=292k29=29⇒k=12⇒a¯=b¯=12(2i¯+3j¯+4k¯)(a¯+b¯)⋅(−7i¯+2j¯+3k¯)=(2i¯+3j¯+4k¯)⋅(−7i¯+2j¯+3k¯)(a¯+b¯)⋅(−7i¯+2j¯+3k¯)=−14+6+12=4

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

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Detailed Solution

$\begin{array}{l} | \bar{a} + \bar{b} | = \sqrt{29}, \bar{a} \times (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) = (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) \times \bar{b} \\ \bar{m} \times \bar{n} = \bar{n} \times \bar{p} \Rightarrow \bar{m} = \bar{p} = k \bar{n} then \bar{a} = \bar{b} = k (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) \\ | k (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) + k (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) | = \sqrt{29} \Rightarrow | 2 k (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) | = \sqrt{29} \\ 2 k \sqrt{29} = \sqrt{29} \Rightarrow k = \frac{1}{2} \Rightarrow \bar{a} = \bar{b} = \frac{1}{2} (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) \\ \begin{aligned} (\bar{a} + \bar{b}) \cdot (- 7 \bar{i} + 2 \bar{j} + 3 \bar{k}) & = (2 \bar{i} + 3 \bar{j} + 4 \bar{k}) \cdot (- 7 \bar{i} + 2 \bar{j} + 3 \bar{k}) \\ (\bar{a} + \bar{b}) \cdot (- 7 \bar{i} + 2 \bar{j} + 3 \bar{k}) & = - 14 + 6 + 12 = 4 \end{aligned} \end{array}$