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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra5 Marks
Question
Show that each of the given three vectors is a unit vector:
$ \frac{1}{7}\left( {2\hat i + 3\hat j + 6\hat k} \right), \frac{1}{7}\left( {6\hat i + 2\hat j - 3\hat k} \right), \ \frac{1}{7}\left( {3\hat i - 6\hat j + 2\hat k} \right)$
Also, show that they are mutually perpendicular to each other.
✓

Answer

$\left| {\vec a} \right| =\frac{1}{7}\sqrt{36+4+9}=\frac{1}{7}\sqrt{49}= 1$

$\left| {\vec b} \right|=\frac{1}{7}\sqrt{36+4+9}=\frac{1}{7}\sqrt{49} = 1$

$\left| {\vec c} \right|=\frac{1}{7}\sqrt{9+36+4}=\frac{1}{7}\sqrt{49} = 1$

Hence they are unit vectors

$\vec a.\vec b = \frac{1}{49}(2\hat i+3\hat j+6\hat k)(6\hat i+2\hat j-3\hat k)$

$=\frac{1}{49}(12+6-18)=0$

$\vec b.\vec c=\frac{1}{49}(6\hat i+2\hat j-3\hat k)(3\hat i-6\hat j+2\hat k)$

$= \frac{1}{49}(18-12-6)=0$

$\vec c.\vec a = \frac{1}{49}(3\vec i-6\vec j+2\vec k)(2\vec i+3\vec j+6\vec k)$

$=\frac{1}{49}(6-18+12)=0$

$\vec a \bot \vec b, \ \vec b \bot \vec c$ and $\vec c \bot \vec a$

So they are $ \bot $ to each other.

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