The sum of two vectors given magnitudes and included angle

[Aaron H.]

Homework Statement

Given the magnitudes of vectors u and v and the angle θ between them, find sum of u + v. Give the magnitude to the nearest tenth when necessary and give the direction by specifying the angle that the resultant makes with u to the nearest degree.

Homework Equations

|u| = 15, |v| = 15, θ = 116°

The Attempt at a Solution

Knowing only the answer (15.9, 58°) and some trig ideas:

I draw an angle of 116 degrees in the starting point of the trig plane. I drop a line from the angle end side, forming a triangle with a 64 degree angle in quadrant II. The other angles of the triangle are both 58 degrees (116 deg / 2). Both opposite (U) and adjacent (V) sides are 15.

SAS - law of cosines

c^2 = (15)^2 + (15)^2 - 2 (15)(15) cos (64 deg)

c = 15.9

angle = 58 deg

(15.9, 58 deg)


What is the correct method?

 

[Dick]

I think that is the correct method. Well done! Draw the quadrilateral that the two vectors and the sum make and use trig.

 

[Aaron H.]

Can you verify this?

|u| = 6, |v| = 11, θ = 42

I drew the quadrilateral. From this, 138 deg captures (u + v). Law of cosines using 6, 11, and 138 deg. c = u + v = 15.97. Round to 16. 42 bisected = 21. Calculated answer (16, 21 deg), however, the answer key presents (16, 27 deg). I have no idea how that angle can be 27 so that must be a mistake.

 

[Dick]

Can you verify this?

|u| = 6, |v| = 11, θ = 42

I drew the quadrilateral. From this, 138 deg captures (u + v). Law of cosines using 6, 11, and 138 deg. c = u + v = 15.97. Round to 16. 42 bisected = 21. Calculated answer (16, 21 deg), however, the answer key presents (16, 27 deg). I have no idea how that angle can be 27 so that must be a mistake.

In your last example you had |u|=|v| so you could just bisect the angle. Here |u| and |v| are different. So the angle of u+v won't bisect the angle. You'll have to do a little more trig to find the right angle. You'll want to find the angle between the sides that are 6 and 16 in the triangle whose sides are 6, 11 and 16.

 

[Aaron H.]

Got it, thanks. The angles of the 6-11-16 triangle are 138 deg, 15 deg, and 27 deg. I used the law of cosines. 27 deg is the angle between u and u + v.

 

[daveb]

As written there are an infinite number of answers since u + v is a vector sum. Are you sure it isn't |u + v|?