High School: Number and Quantity
Vector and Matrix Quantities HSN-VM.A.2
2. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
As we already know, a vector has both direction and magnitude—position and length. Think of the pitcher's mound as the origin. The pitcher can throw in lots of different directions and lots of different distances depending on where the runner is.
The easiest form of a vector is when its initial point is at the origin. Students should know that a vector like this is said to be in standard form. It's standard because those zeros are very easy to work with. (Unless you put them in the denominators of fractions. That's a sin.)
The components of a vector are the differences in the x and y coordinates of its terminal and initial points.
For example, let's say we have a vector v with initial point (2, 3) and terminal point (3, 5). Its components are the difference in the x values (3 – 2 = 1) and the difference in the y values (5 – 3 = 2). That means we could write our vector in component form as v = <1, 2>.
The two notations are equivalent, since a vector from (2, 3) to (3, 5) is equivalent (by definition, remember?) to a standard vector that starts at the origin and ends at (1, 2). We give these vectors the components <1, 2>, and we use the < and > symbols instead of parenthesis to distinguish them from looking like points.