Question

2 vectors a and b are at an angle of 60 with each other the resultant make 45 degree with a. b= 2 units. a=?

Answer 1

The vectors make 60 degrees with each other, and the resultant vector  makes 45 degrees with a

 

The other angle = 15 degrees

 

Using the Sine Rule:

 

a/sin15 = 2/sin45

 

a = 2Sin15/Sin45

 

a = 0.732

Answer 2

The value of a is ($\sqrt{3} -1$) units.

Step-by-step explanation:

We are given that vectors a and b are at an angle of 60° with each other and the resultant makes an angle of 45° with vector a.

As we know that the resultant makes an angle ($\alpha$) with the vector a is given by the following formula;

$\text{tan}(\alpha) = \frac{b \times sin \theta }{a + (b\times cos \theta)}$

Here $\alpha$ = 45° and $\theta$ = 60°

So,  $\text{tan }45\° = \frac{b \times sin60\° }{a + (b\times cos 60\°)}$

         $1 = \frac{b \times \frac{\sqrt{3} }{2} }{a + (b\times \frac{1}{2} )}$        {$\because \text{tan }45\° = 1$ , $\text{sin }60\° = \frac{\sqrt{3} }{2}$  and $\text{cos }60\° = \frac{1 }{2}$ }

So,  ${a + ( \frac{b}{2} )= \frac{\sqrt{3} b}{2} }}$

       ${a + ( \frac{2}{2} )= \frac{\sqrt{3} \times 2}{2} }}$    {as b = 2 units}

        a + 1 = $\sqrt{3}$

        a = ($\sqrt{3} -1$) units

Hence, the value of vector a = ($\sqrt{3} -1$) units.