Answer:
The between the given vector and the positive direction of the x-axis is approximately 41.186º.
Step-by-step explanation:
Let be [tex]\vec u = 8\,i + 7\,j[/tex] and [tex]\vec v = i[/tex] (Positive direction of the x-axis), the angle between both vectors can be determined from definition of dot product:
[tex]\vec u \bullet \vec v = \|\vec u\|\cdot \|\vec v\| \cdot \cos \theta[/tex]
Angle is cleared:
[tex]\cos \theta = \frac{\vec u\bullet \vec v}{\|\vec u\|\cdot \|\vec v\|}[/tex]
[tex]\theta = \cos^{-1}\left(\frac{\vec u \bullet \vec v}{\|\vec u\|\cdot \|\vec v\|} \right)[/tex]
The norms of [tex]\vec u[/tex] and [tex]\vec v[/tex] are found by Pythagorean Theorem:
[tex]\|\vec u\| = \sqrt{8^{2}+7^{2}}[/tex]
[tex]\|\vec u\| \approx \sqrt{113}[/tex]
[tex]\|\vec v\| = 1[/tex]
The dot product between both vectors is:
[tex]\vec u \bullet \vec v = (8)\cdot (1) + (6)\cdot (0)[/tex]
[tex]\vec u \bullet \vec v = 8[/tex]
The angle is now calculated:
[tex]\theta = \cos^{-1}\left(\frac{8}{\sqrt{113}} \right)[/tex]
[tex]\theta \approx 41.186^{\circ}[/tex]
The between the given vector and the positive direction of the x-axis is approximately 41.186º.
