Question

find the magnitude and direction angle (in degrees) of the following vector.

$$ mathbf { v } = 8 ( cos ( 76 ^ { circ } ) mathbf { i } + sin ( 76 ^ { circ } ) mathbf { j } ) $$

Explanation:

Step1: Find the magnitude

The vector is given in the form \(v = r(\cos\theta i+\sin\theta j)\). For a vector \(v = r(\cos\theta i+\sin\theta j)\), the magnitude \(||v||\) is given by \(r\).
Here, \(r = 8\), so \(||v||=8\).

Step2: Find the direction angle

The direction angle \(\theta\) of the vector \(v = r(\cos\theta i+\sin\theta j)\) is the angle inside the cosine and sine functions.
Since the vector is \(v = 8(\cos(76^{\circ})i+\sin(76^{\circ})j)\), the direction angle \(\theta = 76^{\circ}\)

Answer:

The magnitude of the vector is \(8\) and the direction angle is \(76^{\circ}\)