Question

Three vectors in a plane are, respectively, 6, 5, and 4 units long. The first and second form an angle 50°, while the second and third form an angle of 75°. Find the magnitude and direction of the resultant with respect to the larger vector?

Answer 1

See the explanation section please.

Explanation:

Suppose, `vec a = 6,vec b = 5 and vec c =4`

So, if resultant of `vec b and vecc` will be `vec d`

Then, `|vec d| = sqrt(5^2+4^2+2*5*4*cos 75)`

i.e `7.17`

So,angle formed between `vec d and vec b` is

`theta = tan^-1((4 sin 75)/(5+4 cos 75))` i.e `32.62` degrees

So, `vec d` now makes an angle of `(50+32.62)` or `82.62` degrees w.r.t `vec a`

So,if the resultant of `vec a` and `vec d` be `vec e`,then

`|vec e| = sqrt(6^2+7.17^2+2*6*7.17*cos 82.62)`

i.e `9.92`

So,if this `vec e` makes an angle of `mu` w.r.t `vec a`,

then, `mu = tan^-1((7.17 sin 82.62)/(6+7.17 cos 82.62))` i.e `45.77`