Answer:
Magnitude: [tex]\|\vec v\| = 5\sqrt{6}[/tex], Direction (unitized): [tex]\vec u = \frac{\sqrt{6}}{6}\,\hat{i} + \frac{11\sqrt{6}}{12}\,\hat{j}+\frac{5\sqrt{6}}{12}\,\hat{k}[/tex]
Explanation:
Let [tex]\vec v = a\,\hat{i}+b\,\hat{j}+c\,\hat{k}[/tex]. The magnitude of the vector is represented by the Pythagorean formula:
[tex]\|\vec v\| = \sqrt{a^{2}+b^{2}+c^{3}}[/tex] (1)
And the direction is represented by the direction cosines, measured in sexagesimal degrees, that is:
[tex]\theta_{x} = \cos^{-1} \frac{a}{\|\vec v\|}[/tex] (2)
[tex]\theta_{y} = \cos^{-1} \frac{b}{\|\vec v\|}[/tex] (3)
[tex]\theta_{z} = \cos^{-1}\frac{c}{\|\vec v\|}[/tex] (4)
If we know that [tex]a = 2[/tex], [tex]b = 11[/tex] and [tex]c = 5[/tex], then the magnitude and directions of the vector are, respectively:
[tex]\|\vec v\| = \sqrt{2^{2}+11^{2}+5^{2}}[/tex]
[tex]\|\vec v\| = 5\sqrt{6}[/tex]
The direction can be represented by the following unit vector:
[tex]\vec u = \frac{\vec v}{\|\vec v\|}[/tex] (5)
[tex]\vec u = \frac{2\,\hat{i}+11\,\hat{j}+5\,\hat{k}}{2\sqrt{6}}[/tex]
[tex]\vec u = \frac{\sqrt{6}}{6}\,\hat{i} + \frac{11\sqrt{6}}{12}\,\hat{j}+\frac{5\sqrt{6}}{12}\,\hat{k}[/tex]
