1.2 Vectors, components

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A scalar is a quantity with magnitude (i.e. particular power) but no direction.

For instance, a scalar could be mass 65kg or energy 100J. I’m not saying ’65kg down’ or ’100J east’. Get the point?

A vector is essentially an arrow. It points in a direction, and it has a particular power. For instance, a vector could be an arrow 30m North, or 20m South. That is, I’ve also given a direction !

The representation of a vector is by an arrow, revealing the direction, i.e.:

So the vector points from the tail to the head, the direction revealing the direction, and the length of the arrow revealing the magnitude.

For instance, a vector could be force 500N down, or momentum 550sN up.

Sometimes it is necessary to resolve vectors into their components. To resolve vectors into its components, you simply create a vector sum by 2 vectors (horizontal and vertical) 90 degrees to each other, and which whose sum is the original vector.

For instance, instead of the vector, we can resolve it into its 2 black horizontal and vertical components.

Vector resolution is particularly useful because when summing vectors, you can add the x-components and the y-components together, then add everything back together, rather than using complex geometric functions. For example, writing a vector in the format of [x-component, y-component, z-component], to add [1,2,3] and [5,5,5], this is [1+5,2+5,3+5]=[6,7,8], not requiring any complex geometry!

The question then, is obviously how you can find component lengths. Vector resolution component lengths can be found through the Pythagorean theorem and SOH-CAH-TOA enlisted as follows:

An example of SOH-CAH-TOA and the Pythagorean theorem can be found in the following section.

More info: http://phys23p.sl.psu.edu/phys_anim/vectors/embederQ7.102.html