Vector Addition and Subtraction

Pre-Calculus · Axiom Academy

EXAMPLE Vector Addition and Subtraction Combine vectors component-wise: scale each one, then subtract — all from a single rule. Let and . Evaluate the linear combination , expressing the result in component form . Each vector is drawn from the origin. The dashed arrows are the scaled vectors 2u and 3v; the bold orange arrow is the resultant 2u − 3v. Nice work — you scaled two vectors and combined them, which is exactly how every vector sum, difference, and scalar multiple is built. Scalar multiplication: multiply each component by the scalar, e.g. . Order of operations: scale the vectors first, then add or subtract — just like numbers, multiplication comes before subtraction. Add and subtract component-wise: a difference is just adding the negative, , done one coordinate at a time. One rule — operate on each component separately — covers addition, subtraction, and scalar multiplication, and it scales to any linear combination you will meet.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.