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Vector or Cross Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product3 Marks
Question
If $\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},$ and $\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}},$ then find $\vec{\text{a}}\times\vec{\text{b}}.$ verify that $\vec{\text{a}}$ and $\vec{\text{a}}\times\vec{\text{b}}$ are perpendicular to each other.
✓

Answer

Given:
$​\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}$
$​\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}$
$\vec{\text{a}}\times\vec{{\text{b}}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&-2&3\\2&3&-5 \end{vmatrix}$
$=\hat{\text{i}}+11\hat{\text{j}}+7\hat{\text{k}}$
Now,
$\vec{\text{a}}.\big(\vec{\text{a}}\times\vec{\text{b}}\big)=1-22+21$
$=0$
Thus, $\vec{\text{a}}$ is perpendicular to $\vec{\text{a}}\times\vec{\text{b}}.$

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