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:
[tex]\theta = \cos^{-1}\frac{\vec u \,\bullet \,\vec v}{\|\vec u\|\cdot \|\vec v\|}[/tex] (1)
Where:
[tex]\vec u[/tex], [tex]\vec v[/tex] - Vectors, no unit.
[tex]\|\vec u\|[/tex], [tex]\|\vec v\|[/tex] - Norms of vectors, no unit.
[tex]\theta[/tex] - Angle, measured in sexagesimal degrees.
Please notice that norms are calculated by Pythagorean Theorem. If we know that [tex]\vec u = (3,5)[/tex] and [tex]\vec v = (1, 0)[/tex], then the angle between the vector and positive x-axis is:
[tex]\|\vec u\| = \sqrt{3^{2}+5^{2}}[/tex]
[tex]\|\vec u\| = \sqrt{34}[/tex]
[tex]\|\vec v\| = 1[/tex]
[tex]\theta = \cos^{-1}\frac{(3)\cdot (1)+(5)\cdot (0)}{\sqrt{34}\cdot 1 }[/tex]
[tex]\theta = \cos^{-1}\frac{3}{\sqrt{34}}[/tex]
[tex]\theta \approx 59.036^{\circ}[/tex]
The angle between the vector and positive x-axis is approximately 59.036º.
