Question

Two vectors acting through a point are in the ratio 3 : 5. If the angle between them is 60^(@) and the magnitude of their resultant is 35 , find the magnitude of vectors.

Answer 1

Given ;

Ratio of two vectors acting through a point :

=3:5

Angle between the two vectors :

= 60°

Magnitude of their resultant :

=35

To Find :

Magnitude of two vectors = ?

Solution :

Let the magnitude of one of the vectors be :

= 3x

Magnitude of the other vector :

= 5x

∴We know that the formula for the resultant R of any two vectors a and b having any angle $\theta$ between them is :

$R=\sqrt{a^{2} +b^{2} +2abcos\theta }$

So , putting the given values in above equation :

$35= \sqrt{9x^{2}+25x^{2} +2\times 3x\times 5x\times cos60 } \\\\35 =\sqrt{34x^{2} +2\times 15x^{2} \times\frac{1}{2} } \\\\35=\sqrt{49x^{2} } \\7x = 35\\x = 5$

So the magnitude of the vectors are  15 and 25.

Answer 2

Using cosine law and assuming the vectors to be of magnitudes 3x and 5x,

$c = a {}^{2} + b {}^{2} + 2ab \cos(theta)$

$34x {}^{2} + 15x{}^{2}$

therefore x=35/7=5

Forces are thus equal to 15 N and 25 N.