Question

a resultant of two vectors makes 30 ° with one vector and 45° with the other. Find the resultant vectors if the resultant has the magnitude 15.
Hint: law of sines​

Answer 1

Answer:

The correct answers to the question are \ 10\sqrt{3} 10

3

and \ 15\sqrt{2} 15

2

respectively.

CALCULATION:

Let two vectors are denoted as \vec A \ and\ \vec B

A

and

B

Let the resultant of these two vectors is denoted as \vec R

R

.

The angle made by the resultant with A is 30 degree and with B is 45 degree.

We are asked to determine these two vectors.

The angle made by \vec R

R

with \vec A

A

is 30 degree.

The component of \vec A

A

along the direction of \vec R

R

is calculated as - \vec A cos\theta

A

cosθ

Hence, the magnitude of \vec R

R

= Acos\thetaAcosθ

= A×cos30

= A\times \frac{\sqrt{3}}{2}A×

2

3

⇒ A=\ 15\times\frac{2}{\sqrt{3}}A= 15×

3

2

⇒ A=\ 10\sqrt{3}A= 10

3

[ANS]

Similarly the angle made by \vec R

R

with \vec B

B

is 45 degree.

Hence, the component of B along R is \RR = Bcos\thetaBcosθ

= B\times cos45B×cos45

=B\times \frac{1}{ \sqrt{2}}=B×

2

1

⇒ B=\ R\times \sqrt{2}B= R×

2

=\ 15\sqrt{2}= 15

2

[ans]