Explain vector and rectangular components of a vector?

A vector is a quantity with both direction and magnitude. The rectangular components of a vector v refer to vectors along the x and y axes that add up to v.

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A vector has the potential to be an extremely abstract thing, but if you’re just learning about them for the first time, you can simplify things by imaging vectors as arrows:

A vector is a quantity with direction and magnitude. The vector we want to figure out is (4,3) which is shown in purple. What are its direction and magnitude?

Magnitude is pretty easy. It’s just the length of the vector. We can use the pythagorean theorem:

##m=sqrt(x^2+y^2)=5##

So the magnitude of the purple vector is 5. The direction is, using trigonometry,

##theta=tan^-1(4/3)~~53.1^o##

So that’s about 53 degrees from the y-axis. Not too shabby.

The rectangular components of a vector are those two vectors in red and blue. They’re the vectors in the x and y directions that, when added, make the vector you want. Because they’re just in the x and y directions, their magnitudes are super easy to find. Just look at their coordinates. The blue one is (0,3) and had magnitude 3. The red one is (4,0) with magnitude 4.

Vectors can represent all sorts of things. The easiest example is displacement, which is denoted by the letter s and tells you where on the map you are. If the vector above was displacement, we’d be 3 units North and 4 units East of the origin. If it was velocity instead, it would tell us how fast we’re going, and in what direction. We’d be going at 5 units/second in a direction 53 degrees East of due North.

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