Question

Consider the vector below. Determine its length and direction. Vector: 2i+11j+5k2i+11j+5k Magnitude: Direction (unitized):

Answer 1

Answer:

Magnitude: $\|\vec v\| = 5\sqrt{6}$, Direction (unitized): $\vec u = \frac{\sqrt{6}}{6}\,\hat{i} + \frac{11\sqrt{6}}{12}\,\hat{j}+\frac{5\sqrt{6}}{12}\,\hat{k}$

Explanation:

Let $\vec v = a\,\hat{i}+b\,\hat{j}+c\,\hat{k}$. The magnitude of the vector is represented by the Pythagorean formula:

$\|\vec v\| = \sqrt{a^{2}+b^{2}+c^{3}}$ (1)

And the direction is represented by the direction cosines, measured in sexagesimal degrees, that is:

$\theta_{x} = \cos^{-1} \frac{a}{\|\vec v\|}$ (2)

$\theta_{y} = \cos^{-1} \frac{b}{\|\vec v\|}$ (3)

$\theta_{z} = \cos^{-1}\frac{c}{\|\vec v\|}$ (4)

If we know that $a = 2$, $b = 11$ and $c = 5$, then the magnitude and directions of the vector are, respectively:

$\|\vec v\| = \sqrt{2^{2}+11^{2}+5^{2}}$

$\|\vec v\| = 5\sqrt{6}$

The direction can be represented by the following unit vector:

$\vec u = \frac{\vec v}{\|\vec v\|}$ (5)

$\vec u = \frac{2\,\hat{i}+11\,\hat{j}+5\,\hat{k}}{2\sqrt{6}}$

$\vec u = \frac{\sqrt{6}}{6}\,\hat{i} + \frac{11\sqrt{6}}{12}\,\hat{j}+\frac{5\sqrt{6}}{12}\,\hat{k}$