Points, Vectors, and Functions
When Juliet said "that which we call a rose / by any other name would smell as sweet" she could have been talking about math.
As Romeo would retain his "dear perfection" if he were called something else, a vector retains its x and y components regardless of how we write the vector.
Vectors are much less complicated than people. We've never heard of anyone dying for vector love.
The purpose of vector notation is to keep the x and y components straight. Here are some of the many ways we can write a vector whose x-component is 5 and whose y-component is 4.
- We can use pointy brackets. The x-component comes first and the y-component second:
- We can use bold letters to denote the components. The coefficient in front of the letter i is the x-component and the coefficient in front of the letter j is the y-component:
5i + 4j
- We can use a matrix where the upper entry is the x-component and the lower entry is the y-component:
- We can use parentheses instead of pointy brackets, although this notation can lead to confusion between vectors and points:
- When we see this notation, we need to be careful to discern the difference between a vector and a point. Remember a 2-D point is an ordered pair of coordinates. A 2-D vector is represented by two coordinates that denote the vectors magnitude and direction.