Two vectors having magnitude 12 and 13 are inclined at an angle 45 degree

Question

with each other.find their resultant vector...how to find that?

Answer 1

Magnitude of two vectors |a| = 12 and |b| = 13

Find the magnitude of resultant vector by using formula

R = √[a² + b²  + 2ab cos(θ)]

R = √[12²  + 13²  + 2(12)(13) cos(45)]

= √[144  + 169  + 312 (1/√2)]

= √(313  + 220.65)

= √(533.65)

R = 23.10

Find the direction of resultant vector by using formula α = tan^-1 [ b sin(θ)/(a + b cos(θ)]

α = tan^-1 [ 13 sin(45)/(12 + 13 cos(45)]

= tan^-1 { [ 13(1/√2)]/[12 + 13(1/√2)] }

= tan^-1 {(13/√2)/[12 + (13/√2)]}

= tan^-1 [(13/√2)/[(12√2 +13)/√2)]

= tan^-1 [13/(12√2 +13)]

= tan^-1 [13/(16.968 +13)]

= tan^-1 (13/29.968)

= tan^-1 (0.433)

α = 23.41^o.