High School: Number and Quantity
Vector and Matrix Quantities HSN-VM.B.4a
a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
Students should know that to add two vectors geometrically, we position them so that the initial point of one is the terminal point of the other. In other words, attach the start of one to the end of the other. One of two things will happen: either they'll form a straight line (if their directions are the same) or they'll form an angle.
When the vectors don't form a straight line, students should know that we connect the two unattached points of the vectors, and ta-da! We'll have the sum. This is what's called parallelogram rule. (That's because the vector u + v is the diagonal of a parallelogram whose adjacent sides are u and v.)
Students should also know that component form means we're given the horizontal and vertical distances a vector covers. To add vectors in component form, all we need to do is add their components. These components can also be used to calculate the direction of the vectors' sum.