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Scalar Or Dot Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
If $\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then show that the vectores $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}$ are orthonal.
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Answer

Given that
$\vec{\text{a}}=5\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}};\vec{\text{b}}=\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}$
$\therefore\vec{\text{a}}+\vec{\text{b}}=5\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}+\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}=6\hat{\text{i}}+2\hat{\text{j}}-8\hat{\text{k}}$
And $\vec{\text{a}}-\vec{\text{b}}=5\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}-\big(\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}\big)=4\hat{\text{i}}-4\hat{\text{j}}+2\hat{\text{k}}$
Now,
$\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)$
$=\big(6\hat{\text{i}}+2\hat{\text{j}}-8\hat{\text{k}}\big).\big(4\hat{\text{i}}-4\hat{\text{j}}+2\hat{\text{k}}\big)$
$=24-8-16$
$=0$
So, $\vec{\text{a}}+\vec{\text{b}}$ is orthogonal to $\vec{\text{a}}-\vec{\text{b}}.$

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