Find the component form of the vector v ⃗ with ‖v ⃗ ‖=4√3 when drawn in standard position v ⃗ lies in Quadrant II and makes a 30° angle with the positive y-axis. Give exact values.

The answer:

first of all, we should know that the expression of a vector V (a, b) can be written as follow:

V = r (Vx i + Vyj), where r is the length of the vector, it is r = sqrt (V²x + V²y)

Vx is the component lying on the x-axis and Vy on the y-axis

v ⃗ lies in Quadrant II, means Vx is less than 0 (negative)

so Vx = - r sin30° and Vy = rcos30°

r = ‖v ⃗ ‖=4√3

so we have v = - 4√3sin30° i + 4√3 cos30° j

the components are

v ( - 4√3sin30°, 4√3 cos30°) = (-2√3, 4√3 cos30°)