# Scalar and Vector Examples

Here you will learn some scalar and vector examples for better understanding of scalar and vector concepts.

Example 1 : Find the vector of magnitude 5 which are perpendicular to the vectors $$\vec{a}$$ = $$2\hat{i} + \hat{j} – 3\hat{k}$$ and $$\vec{b}$$ = $$\hat{i} – 2\hat{j} + \hat{k}$$.

Solution : Unit vectors perpendicular to $$\vec{a}$$ & $$\vec{b}$$ = $$\pm$$$$\vec{a}\times\vec{b}\over |\vec{a}\times\vec{b}|$$

$$\therefore$$   $$\vec{a}\times\vec{b}$$ = $$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -3 \\ 1 & 2 & -2 \\ \end{vmatrix}$$ = $$-5\hat{i} – 5\hat{j} – 5\hat{k}$$

$$\therefore$$   Unit Vectors = $$\pm$$ $$-5\hat{i} – 5\hat{j} – 5\hat{k}\over 5\sqrt{3}$$

Hence the required vectors are $$\pm$$ $$5\sqrt{3}\over 3$$($$\hat{i} + \hat{j} + \hat{k}$$)

Example 2 : If $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are three non zero vectors such that $$\vec{a}\times\vec{b}$$ = $$\vec{c}$$ and $$\vec{b}\times\vec{c}$$ = $$\vec{a}$$, prove that $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are mutually at right angles and |$$\vec{b}$$| = 1 and |$$\vec{c}$$| = |$$\vec{a}$$|

Solution : $$\vec{a}\times\vec{b}$$ = $$\vec{c}$$ and $$\vec{b}\times\vec{c}$$ = $$\vec{a}$$

$$\implies$$   $$\vec{c}\perp\vec{a}$$ , $$\vec{c}\perp\vec{b}$$ and $$\vec{a}\perp\vec{b}$$, $$\vec{a}\perp\vec{c}$$

$$\implies$$   $$\vec{a}\perp\vec{b}$$, $$\vec{b}\perp\vec{c}$$ and $$\vec{c}\perp\vec{a}$$

$$\implies$$   $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ are mutually perpendicular vectors.

Again, $$\vec{a}\times\vec{b}$$ = $$\vec{c}$$ and $$\vec{b}\times\vec{c}$$ = $$\vec{a}$$

$$\implies$$   |$$\vec{a}\times\vec{b}$$| = |$$\vec{c}$$| and |$$\vec{b}\times\vec{c}$$| = |$$\vec{a}$$|

$$\implies$$   $$|\vec{a}||\vec{b}|sin{\pi\over 2}$$ = |$$\vec{c}$$| and $$|\vec{b}||\vec{c}|sin{\pi\over 2}$$ = |$$\vec{a}$$|     ($$\because$$ $$\vec{a}\perp\vec{b}$$ and $$\vec{b}\perp\vec{c}$$)

$$\implies$$   $$|\vec{a}||\vec{b}|$$ = |$$\vec{c}$$| and $$|\vec{b}||\vec{c}|$$ = |$$\vec{a}$$|

$$\implies$$   $${|\vec{b}|}^2$$ |$$\vec{c}$$| = |$$\vec{c}$$|

$$\implies$$   $${|\vec{b}|}^2$$ = 1

$$\implies$$   $$|\vec{b}|$$ = 1

putting in $$|\vec{a}||\vec{b}|$$ = |$$\vec{c}$$|

$$\implies$$   $$|\vec{a}|$$ = |$$\vec{c}$$|

Example 3 : For any three vectors $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ prove that [$$\vec{a}$$ + $$\vec{b}$$ $$\vec{b}$$ + $$\vec{c}$$ $$\vec{c}$$ + $$\vec{a}$$] = 2[$$\vec{a}$$ $$\vec{b}$$ $$\vec{c}$$]

Solution : We have [$$\vec{a}$$ + $$\vec{b}$$ $$\vec{b}$$ + $$\vec{c}$$ $$\vec{c}$$ + $$\vec{a}$$]

= {($$\vec{a}$$ + $$\vec{b}$$)$$\times$$($$\vec{b}$$ + $$\vec{c}$$)}.($$\vec{c}$$ + $$\vec{a}$$)

= {$$\vec{a}$$$$\times$$$$\vec{b}$$ + $$\vec{a}$$$$\times$$$$\vec{c}$$ + $$\vec{b}$$$$\times$$$$\vec{b}$$ + $$\vec{b}$$$$\times$$$$\vec{c}$$}.($$\vec{c}$$ + $$\vec{a}$$)       {$$\vec{b}$$$$\times$$$$\vec{b}$$ = 0}

= {$$\vec{a}$$$$\times$$$$\vec{b}$$ + $$\vec{a}$$$$\times$$$$\vec{c}$$ + $$\vec{b}$$$$\times$$$$\vec{c}$$}.($$\vec{c}$$ + $$\vec{a}$$)

= ($$\vec{a}\times\vec{b}$$).$$\vec{c}$$ + ($$\vec{a}\times\vec{c}$$).$$\vec{c}$$ + ($$\vec{b}\times\vec{c}$$).$$\vec{c}$$ + ($$\vec{a}\times\vec{b}$$).$$\vec{a}$$ + ($$\vec{a}\times\vec{c}$$).$$\vec{a}$$ + ($$\vec{b}\times\vec{c}$$).$$\vec{a}$$

= [$$\vec{a}$$ $$\vec{b}$$ $$\vec{c}$$] + 0 + 0 + 0 + 0 + [$$\vec{b}$$ $$\vec{c}$$ $$\vec{a}$$]

= [$$\vec{a}$$ $$\vec{b}$$ $$\vec{c}$$] + [$$\vec{a}$$ $$\vec{b}$$ $$\vec{c}$$] = 2[$$\vec{a}$$ $$\vec{b}$$ $$\vec{c}$$]

Practice these given scalar and vector examples to test your knowledge on concepts of scalar and vectors.