Engineering Corner

Vector Mathematical Operations

Vectors can be manipulated mathematically in various ways to describe physical quantities and their interactions. Common operations include:

1. Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation changes the magnitude of the vector while maintaining its direction, unless the scalar is negative, in which case the direction is reversed.

Multiply by 2

Doubles the vector's magnitude

Multiply by -1

Reverses the vector's direction.

Divide by 2

Reduces the vector's magnitude to half (like multiplying by 1/2).

2. Vector Addition

Vector addition combines two or more vectors to produce a resultant vector.

Vectors are added tip-to-tail - Place the tail of one vector at the tip of the other, then draw the resultant vector from the tail of the first to the tip of the last.

Vector addition

The triangle method for vector addition demonstrates that the order of addition does not matter because vector addition is commutative —the order in which vectors are added does not affect the resultant..

3. Vector Subtraction

Vector subtraction is a special case of vector addition. Instead of using the tip-to-tail method, the vectors are placed tail-to-tail, and the resultant vector is drawn from the tail of the first vector to where the heads of the two vectors would meet.

Looking closely, it is possible to see that this is equivalent to adding the second vector in the negative direction. Using the tip-to-tail method, A + (-B) will produce the same result so A - B = A + (-B).

Interactive Vector Addition

Adjust the vector components and see the resultant vector directly on the graph.

Vector Inputs

A-x: A-y: Color A:
B-x: B-y: Color B:

Axes Settings

X Min: X Max:

Y Min: Y Max: