Question

Engineering Physics
The x and y component of a vector is 3 and 5, respectively. What is the angle between the vector and positive x-axis?

Answer 1

Answer:

The angle between the vector and positive x-axis is approximately 59.036º.

Explanation:

By Linear Algebra and to be precise, by definition of Dot Product we can determine the angle between two vector from following expression:

$\theta = \cos^{-1}\frac{\vec u \,\bullet \,\vec v}{\|\vec u\|\cdot \|\vec v\|}$ (1)

Where:

$\vec u$, $\vec v$ - Vectors, no unit.

$\|\vec u\|$, $\|\vec v\|$ - Norms of vectors, no unit.

$\theta$ - Angle, measured in sexagesimal degrees.

Please notice that norms are calculated by Pythagorean Theorem. If we know that $\vec u = (3,5)$ and $\vec v = (1, 0)$, then the angle between the vector and positive x-axis is:

$\|\vec u\| = \sqrt{3^{2}+5^{2}}$

$\|\vec u\| = \sqrt{34}$

$\|\vec v\| = 1$

$\theta = \cos^{-1}\frac{(3)\cdot (1)+(5)\cdot (0)}{\sqrt{34}\cdot 1 }$

$\theta = \cos^{-1}\frac{3}{\sqrt{34}}$

$\theta \approx 59.036^{\circ}$

The angle between the vector and positive x-axis is approximately 59.036º.