Question

Add vectors →A, →B and →C each having magnitude of 100 unit and inclined to the X-axis at angles 45°, 135° and 315° respectively.

Answer 1

The resultant vector’s magnitude is 100 units and at x-axis, an angle of 45° is made.

Explanation:

First, along the x-axis and y-axis, the vector’s components should be found. Then, the resultant y and x-components can be found.

x-component of A→ = Acos45° = 100 cos 45° = 100/√2 unit

x-component of B→ = B→ cos 135° = - 100/√2

x-component of C→ = C→ cos 315° = 100 cos 315° = 100 cos 45°= 100/√2

x-component’s resultant = $\frac{100}{\sqrt{2} } + \frac{100}{\sqrt{2} } - \frac{100}{\sqrt{2} } = \frac{100}{\sqrt{2} }$

Now, y-component of A→ = 100 sin 45° = $\frac{100}{\sqrt{2} }$

y-component of B→ = 100 sin 135° = 100/√2

y-component of C→ = 100 sin 315° = -100/√2

y-component’s resultant =$\frac{100}{\sqrt{2} } - \frac{100}{\sqrt{2} } + \frac{100}{\sqrt{2} } = \frac{100}{\sqrt{2} }$

The resultant vector makes an angle with the x-axis and it is shown as  

tan α = component of y/component of x      

$=\frac{100/\sqrt{2} }{100/\sqrt{2} } =1$

⇒ α = tan−1 (1)

= 45°