Question

Find the magnitude and direction angle of the vector v = 3(cos60i + sin60j).

a) Magnitude: 3, Direction Angle: 60°
b) Magnitude: 3, Direction Angle: 30°
c) Magnitude: 3√3, Direction Angle: 60°
d) Magnitude: 3√3, Direction Angle: 30°

Answer 1

The magnitude of the vector v = 3(cos60i + sin60j) is 3√3, and the direction angle is 60°, which corresponds to option (c).

Step-by-step explanation:

To find the magnitude and direction angle of the vector v = 3(cos60i + sin60j), we start by evaluating the cosine and sine functions. Since cos60° = 0.5 and sin60° = √3/2, the vector can be rewritten as v = 3(0.5i + √3/2j), which simplifies to v = 1.5i + 2.598j.

To compute the magnitude of the vector, we use the Pythagorean theorem:

Magnitude = √(1.5^2 + 2.598^2) = √ (2.25 + 6.7504) = √9 = 3√3

The direction angle, often measured from the positive x-axis, for the vector is the angle whose tangent is the ratio of j-component to i-component. So, we have:

Tan(θ) = 2.598 / 1.5 ⇒ θ = tan^(-1)(2.598 / 1.5)

Using a calculator, this yields:

θ ≈ 60°

Therefore, the correct magnitude is 3√3 and the direction angle is 60°, which corresponds to option (c).