Component Vectors

The Unit Vector

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1 of 3 videos by James S.

Key Questions

• A vector has both magnitude (which is its length) and direction (which is its angle).

Any two dimentional vector at an angle will have a horizontal and a vertical component .
A vector written as ( 12 , 8 ) will have 12 as its horizontal component, and 8 as its vertical component, and because both components are positive, the vector is pointing to the northeastern direction.

• When two forces are combined, one with magnitude ${r}_{1}$ and direction ${z}_{1}$ and the other with magnitude ${r}_{2}$ and direction ${z}_{2}$, the resultant can be decomposed into horizontal and vertical components <$x , y$>, where:

$x = {r}_{1} \cos \left({z}_{1}\right) + {r}_{2} \cos \left({z}_{2}\right)$
$y = {r}_{1} \sin \left({z}_{1}\right) + {r}_{2} \sin \left({z}_{2}\right)$

example: a force of 20 mph at 30 degrees is combined with a force of 50 mph at 40 degrees. The resultant of the combined forces is:

<$x , y$>$=$<$20 \cos \left(30\right) + 50 \cos \left(40\right) , 20 \sin \left(30\right) + 50 \sin \left(40\right)$>$=$ < $55.6 , 45$>

• To find the magnitude of a vector using its components you use Pitagora´s Theorem.

Consider in 2 dimensions a vector $\vec{v}$ given as:
$\vec{v} = 5 \vec{i} + 3 \vec{j}$ (where $\vec{i}$ and $\vec{j}$ are the unit vectors on the x and y axes)

The magnitude of this vector (or its length in geometrical sense) is given using Pitagora's Theorem, as:
$| \vec{v} | = \sqrt{{5}^{2} + {3}^{2}} = 5 , 8$

The same thing applies in 3 dimensions, the only thing is to include the third component.

So if the vector is now given as:
$\vec{v} = 5 \vec{i} + 3 \vec{j} + 2 \vec{k}$
The magnitude will be:
$| \vec{v} | = \sqrt{{5}^{2} + {3}^{2} + {2}^{2}} = 6 , 2$

• Often when two processes interact we only know the component vector values and need to be able to combine these to get a desired result.

This might be more easily understood by an example:
Suppose I am trying to fly from point A to point B which is due North of point A. My plane flies at an air speed of 100 miles/hour but there is a wind blowing due West at 30 miles/hour. How many degrees East of North do I need to orient my plane to fly in a straight line to B?

From the above diagram, I need to head my plane (approximately)
${17.5}^{o}$ East of North.

This problem could be extended to ask:
If it is 200 miles from A to B and my plane has enough gas to fly 250 miles will I be able to make this trip?

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