Question

obtain the magnitude of 2A - 3B vector If A vector = i cap + j cap - 2k cap and B vector = 2i cap - j cap + k cap

Answer 1

Given:

$A = \hat{i} + \hat{j} - 2 \hat{k}$

$B = 2 \hat{i} - \hat{j} + \hat{k}$

To find:

Magnitude of 2A - 3B.

Calculation:

First let's calculate the individual vectors.

$\therefore \: 2A = 2 \: \bigg \{ \hat{i} + \hat{j} - 2 \hat{k} \bigg \}$

$= > \: 2A = 2\hat{i} + 2\hat{j} - 4 \hat{k}$

Again for vector B:

$\therefore \: 3B = 3 \: \bigg \{2 \hat{i} - \hat{j} + \hat{k} \bigg \}$

$= > \: 3B = 6 \hat{i} - 3\hat{j} + 3\hat{k}$

Now, net vector:

$= > \: 2A + 3B= (2\hat{i} + 2\hat{j} - 4 \hat{k}) + (6 \hat{i} - 3\hat{j} + 3\hat{k} )$

$= > \: 2A + 3B= 8\hat{i} - \hat{j} - \hat{k}$

Magnitude of net vector:

$= > \: | 2A + 3B| = \sqrt{ {8}^{2} + {( - 1)}^{2} + {( - 1)}^{2} }$

$= > \: | 2A + 3B| = \sqrt{ 64+ 1 + 1 }$

$= > \: | 2A + 3B| = \sqrt{ 66 }$

So, final answer is :

$\boxed{ \red{ \bold{ \: | 2A + 3B| = \sqrt{ 66 } }}}$

Answer 2

Given:

A vector = i cap + j cap - 2k cap and B vector = 2i cap - j cap + k cap

To find:

obtain the magnitude of 2A - 3B vector If A vector = i cap + j cap - 2k cap and B vector = 2i cap - j cap + k cap

Solution:

From given, we have,

A vector = i cap + j cap - 2k cap and B vector = 2i cap - j cap + k cap

Now consider,

2A = 2 × (i cap + j cap - 2k cap ) = 2i cap + 2j cap - 4k cap

Similarly, now consider,

3B = 3 × (2i cap - j cap + k cap ) = 6i cap - 3j cap + 3k cap

Now we have,

2A - 3B = [2i cap + 2j cap - 4k cap ] - [6i cap - 3j cap + 3k cap]

2A - 3B = (2i cap - 6i cap) + (2j cap + 3j cap) + (- 4k cap - 3k cap)

2A - 3B = - 4i cap + 5j cap - 7k cap

$\therefore 2A-3B=-4 \hat {i} + 5 \hat {j}-7 \hat {k}$

Therefore, 2A - 3B = - 4i cap + 5j cap - 7k cap

Magnitude |2A - 3B| = √[(-4)² + (5)² + (-7)²] = √[16 + 25 + 49] = √90 = 3√10

Therefore, the magnitude of 2A - 3B is 3√10