Question

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

Answer 1

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

Step-by-step 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) (a)/(\|\vec v\|)` (2)

`\theta_(y) = \cos^(-1) (b)/(\|\vec v\|)` (3)

`\theta_(z) = \cos^(-1)(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√(6)`

The direction can be represented by the following unit vector:

`\vec u = (\vec v)/(\|\vec v\|)` (5)

`\vec u = \frac{2\,\hat{i}+11\,\hat{j}+5\,\hat{k}}{2√(6)}`

`\vec u = (√(6))/(6)\,\hat{i} + (11√(6))/(12)\,\hat{j}+(5√(6))/(12)\,\hat{k}`