Question

Given the vector u equal to 7 (cos 310°, sin 310°) and vector v equal to

2 (cos 115°, sin 115°), find the sum u + v and write your answer in magnitude
and direction form with the magnitude rounded to the nearest tenth and the
direction rounded to the nearest degree, 0° ≤ 0 < 360°.

Answer 1

To find the sum of vectors u and v, we can add their components separately:

(u + v)x = (7cos 310° + 2cos 115°)

(u + v)y = (7sin 310° + 2sin 115°)

Using a calculator to evaluate these expressions, we get:

(u + v)x = -2.16

(u + v)y = 8.48

The magnitude of the sum vector u + v can be found using the Pythagorean theorem:

|u + v| = √((u + v)x^2 + (u + v)y^2)

= √(-2.16^2 + 8.48^2)

= √(72.3584)

= 8.51

The direction of the sum vector u + v can be found using the inverse tangent function:

θ = tan^(-1)((u + v)y / (u + v)x)

= tan^(-1)(8.48 / -2.16)

= tan^(-1)(-3.928571428571429)

= 143.13°

Rounding the magnitude to the nearest tenth and the direction to the nearest degree, we get:

u + v = 8.5 (cos 143°, sin 143°)

Therefore, the sum of vectors u and v is equal to 8.5 (cos 143°, sin 143°).