Name: Vectors: Introduction
Uploaded: 2019-07-19
Duration: 6 min 59 s
Description: Physicists, air traffic controllers, and video game creators have at least one thing in common: vectors. But what exactly are they? Let's dive in to learn more about vectors.

Question

If

\bar{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}, \bar{b} = 3 \hat{i} + 2 \hat{j}

and

c = 2 \hat{i} + \hat{j} + 3 \hat{k}

then verify that

\bar{a} \times (\bar{b} \times \bar{c}) = (\bar{a} \cdot \bar{c}) \bar{b} - (\bar{a} \cdot \bar{b}) \bar{c}

Accepted Answer

Given, $\bar{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}, \bar{b} = 3 \hat{i} + 2 \hat{j}$ and $c = 2 \hat{i} + \hat{j} + 3 \hat{k}$

$\bar{b} \times \bar{c} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 2 & 0 \\ 2 & 1 & 3 \end{matrix}|$

$= \hat{i} (6 - 0) - \hat{j} (9 - 0) + \hat{k} (3 - 4)$

$= 6 \hat{i} - 9 \hat{j} - \hat{k}$

$∴ a \times (\bar{b} \times \bar{c}) = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 6 & - 9 & - 1 \end{matrix}|$

$= \hat{i} (- 2 + 27) - \hat{j} (- 1 - 18) + \hat{k} (- 9 - 12)$

$= 25 \hat{i} + 19 \hat{j} - 21 \hat{k}$ ..........(i)

$\bar{a} . \bar{c} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) . (2 \hat{i} + \hat{j} + 3 \hat{k})$

$= (1) (2) + (2) (1) + (3) (3) = 13$

$∴ (\bar{a} . \bar{c}) \bar{b} = 13 (3 \hat{i} + 2 \hat{j}) = 39 \hat{i} + 26 \hat{j}$

Also, $\bar{a} . \bar{b} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) . (3 \hat{i} + 2 \hat{j})$

$= (1) (3) + (2) (2) + 3 (0) = 7$

$∴ (\bar{a} . \bar{b}) \bar{c} = 7 (2 \hat{i} + \hat{j} + 3 \hat{k}) = 14 \hat{i} + 7 \hat{j} + 21 \hat{k}$

Now, $(\bar{a} . \bar{c}) \bar{b} - (\bar{a} . \bar{b}) \bar{c} = 39 \hat{i} + 26 \hat{j} - (14 \hat{i} + 7 \hat{j} + 21 \hat{k})$

$= 25 \hat{i} + 19 \hat{j} - 21 \hat{k}$ ...(ii)

From (i) and (ii), we get

$\bar{a} \times (\bar{b} \times \bar{c}) = (\bar{a} \cdot \bar{c}) \bar{b} - (\bar{a} \cdot \bar{b}) \bar{c}$

If $\bar{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}, \bar{b} = 3 \hat{i} + 2 \hat{j}$ and $c = 2 \hat{i} + \hat{j} + 3 \hat{k}$ then verify that $\bar{a} \times (\bar{b} \times \bar{c}) = (\bar{a} \cdot \bar{c}) \bar{b} - (\bar{a} \cdot \bar{b}) \bar{c}$

Important Questions on Vectors

Learn and Prepare for any exam you want !

If a¯=i^+2j^+3k^, b¯=3i^+2j^ and c=2i^+j^+3k^ then verify that a¯×(b¯×c¯)=(a¯·c¯)b¯-(a¯·b¯)c¯

Important Questions on Vectors

Learn and Prepare for any exam you want !

If $\bar{a} = \hat{i} + 2 \hat{j} + 3 \hat{k}, \bar{b} = 3 \hat{i} + 2 \hat{j}$ and $c = 2 \hat{i} + \hat{j} + 3 \hat{k}$ then verify that $\bar{a} \times (\bar{b} \times \bar{c}) = (\bar{a} \cdot \bar{c}) \bar{b} - (\bar{a} \cdot \bar{b}) \bar{c}$