vectors problems

Question

1. Use geometric addition of vectors to find the resultant of each combination.
2. 18 km/h north and 16 km/h east. 
3. 12 N on a bearing of 25° and 17 N on a bearing of 100°. 
4. Determine the vertical and horizontal components of each vector.
5. 810 N at an incline of 15° from horizontal
6. 80 m/s, 60° clockwise from vertical

Answer 1

Problem 1:

Use the Pythagorean Theorem:

a² + b² = c²

16² + 18² = c²

c = √(16² + 18²) = 24.08

So the magnitude of the resultant vector is 24.08 kn/hr

Find the direction using:

tan(θ) = (16/18)

θ = tan^-1(16/18) = 41.63°

So the direction of the resultant vector is N41.63°E

For the second problem in 1): 12 N at a bearing of 25° means 25° right (CW) of north. Also, a bearing of 100° means 100° right (CW) of north:

So the angle between the vectors is 75°. Since the vectors and their opposite sides form a parallelogram, ∠α is the supplement to 75° so ∠α = 105°. We can then use the Law of Cosines to solve for the magnitude of the resultant vector (R):

R² = 12² + 17² - 2(12)(17)cos(105°) = 538.598

R = √538.598 = 23.2077 or rounded, the magnitude of the resultant is 23.2 N

The direction can be found using the Law of Sines:

23.2077/sin(105°) = 17/sin(β)

sin(β) = 0.707555

β = 45.04°

So the direction would be a bearing of 70.04° (25 + 45.04)

For problem 2), draw a sketch and make a triangle from the horizontal or vertical so there is a right angle in the triangle. Then use sine and cosine to find the vertical and horizontal components of the vectors.