Question

The ratio of magnitude of two forces is 3:5 and the angle between them is 60 degree. If magnitude of their resultant is 35N Find both the Vectors.​

Answer 1

$\\\bold{Let} \\\bold{Magnitude \: 1 = 3x } \\ \bold{Magnitude \: 2 = 5x\:and} \\ \bold{ We \: Have} \\ \bold{\theta = 60 \degree} \\ \bold{Resultant \: \: = 35 \: N} \\ \bold{Now}\\\bold{According\:To\:Question} \\ = > \bold{35 = \sqrt{ {(3x}^{2} + {(5x}^{2} + 2 \times 5x \times 3x \times \cos(60 \degree}) } \\ = > \bold{35 = \sqrt{9 {x}^{2} + 25 {x}^{2} + 30 {x}^{2} \times \frac{1}{2}}} \\ = > \bold{35 = \sqrt{49 {x}^{2} } } \\ = > \bold{35 \times 35 = 49 {x}^{2} } \\ = > \bold {x}^{2} = \bold{\frac{35 \times 35}{49} } \\ = > \bold{ {x}^{2} = 25 } \\ = > \boxed{\bold{x = 5}}$

Therefore

Magnitude 1 = 3x = 3 X 5 = 15

Magnitude 2 = 5x = 5 X 5 = 25

Answer 2

$\underline{\mathfrak{Answer:-}}$

15N and 25N

$\underline{\mathfrak{Explanation:-}}$

Given

ratio of magnitudes of two forces = 3:5

Angle between them (Φ) = 60°

magnitude of resultant vector = 35N

To Find

magnitude of the both vectors

Solution

Let, the magnitude of the two vectors A and B are 3x and 5x respectively

Resultant vector (R) = 35N

Φ = 60°

We know that

$\boxed{r = \sqrt{{a}^{2}+{b}^{2}+2ab cos \theta}}$

on putting the values

$\mathsf{35 = \sqrt{{(3x)}^{2}+{(5x)}^{2}+2(3x)(5x) cos 60 \degree}}$

$\mathsf{35 = \sqrt{9{x}^{2}+25{x}^{2}+2(3x)(5x) (\dfrac{1}{2}) }}$

$\mathsf{35 = \sqrt{34{x}^{2}+15{x}^{2} }}$

$\mathsf{35 = \sqrt{49{x}^{2}}}$

$\mathsf{35 = 7x}$

$\mathsf{x = \dfrac{35}{7}}$

$\boxed{\bold{x = 5}}$

Let us find the magnitude of two vectors

vector A = 3x

A = 3(5)

A = 15N

vector B = 5x

B = 5(5)

B = 25N

Note:

small letters (a, b, r) represents magnitude