Question

Consider the following vectors in component form: a=⟨2,10⟩, b=⟨−5,−12⟩, and c=a−b.

What is the magnitude and direction angle of vector c?

A) ∣c∣=5,θ=36.9°

B) ∣c∣=5,θ=53.1°

C) ∣c∣=25,θ=16.3°

D) ∣c∣=25,θ=73.7°

Answer 1

The magnitude of vector c is approximately 23.09, and its direction angle is approximately 72.34°. The closest option is D) |c| = 25, θ = 73.7°.

Step-by-step explanation:

To solve the mathematical problem completely and find the magnitude and direction angle of vector c = a - b, we first calculate the components of vector c by subtracting the components of vector b from those of vector a:

• cx = ax - bx = 2 - (-5) = 7
• cy = ay - by = 10 - (-12) = 22

Next, we calculate the magnitude of vector c using the Pythagorean theorem:

||c|| = √(cx² + cy²) = √(7² + 22²) = √(49 + 484) = √533 ≈ 23.09

Now, we find the direction angle θ of vector c using the arctangent:

θ = arctan(cy/cx) = arctan(22/7) ≈ 72.34°

Therefore, the magnitude of vector c is approximately 23.09, and its direction angle is approximately 72.34°. The mention correct option in the final answer closest to these values is:

Option D) |c| = 25, θ = 73.7°