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Scalar Or Dot Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
Let $\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}.$ Find $\lambda$ such that $\vec{\text{a}}+\vec{\text{b}}$ is orthonal to $\vec{\text{a}}-\vec{\text{b}}.$
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Answer

Give that
$\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}};\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}$
$\therefore\vec{\text{a}}+\vec{\text{b}}=5\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}+\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}=6\hat{\text{i}}-2\hat{\text{j}}+(7+\lambda)\hat{\text{k}}$
and $\vec{\text{a}}-\vec{\text{b}}=5\hat{\text{i}}-\hat{\text{j}}+7\hat{\text{k}}-\big(\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}\big)=4\hat{\text{i}}+0\hat{\text{j}}+(7-\lambda)\hat{\text{k}}$
Given that $\vec{\text{a}}+\vec{\text{b}}$ is orthogonal to $\vec{\text{a}}-\vec{\text{b}}.$
$\Rightarrow\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=0$
$\Rightarrow\big[6\hat{\text{i}}-2\hat{\text{j}}+(7+\lambda)\hat{\text{k}}\big].\big[4\hat{\text{i}}+0\hat{\text{j}}+(7-\lambda)\hat{\text{k}}\big]=0$
$\Rightarrow24+0+49-\lambda^2=0$
$\Rightarrow\lambda^2=73$
$\Rightarrow\lambda=\sqrt{73}$

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