Physics » Two-Dimensional Kinematics » Vector Addition and Subtraction: Graphical Methods

Summarizing the Graphical Method of Vector Addition and Subtraction

Summary of the Graphical Method of Adding and Subtracting Vectors

• The graphical method of adding vectors $$\mathbf{A}$$ and $$\mathbf{B}$$ involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector $$\mathbf{R}$$ is defined such that $$\mathbf{\text{A}}+\mathbf{\text{B}}=\mathbf{\text{R}}$$. The magnitude and direction of $$\mathbf{R}$$ are then determined with a ruler and protractor, respectively.
• The graphical method of subtracting vectors $$\mathbf{B}$$ from $$\mathbf{A}$$ involves adding the opposite of vector $$\mathbf{B}$$, which is defined as $$-\mathbf{B}$$. In this case, $$\mathbf{A}–\mathbf{\text{B}}=\mathbf{\text{A}}+\left(\mathbf{–B}\right)=\mathbf{R}$$. Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector $$\mathbf{R}$$.
• Addition of vectors is commutative such that $$\mathbf{\text{A}}+\mathbf{\text{B}}=\mathbf{\text{B}}+\mathbf{\text{A}}$$ .
• The head-to-tail method of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
• If a vector $$\mathbf{A}$$ is multiplied by a scalar quantity $$c$$, the magnitude of the product is given by $$\text{cA}$$. If $$c$$ is positive, the direction of the product points in the same direction as $$\mathbf{A}$$; if $$c$$ is negative, the direction of the product points in the opposite direction as $$\mathbf{A}$$.

Glossary

component (of a 2-d vector)

a piece of a vector that points in either the vertical or the horizontal direction; every 2-d vector can be expressed as a sum of two vertical and horizontal vector components

commutative

refers to the interchangeability of order in a function; vector addition is commutative because the order in which vectors are added together does not affect the final sum

direction (of a vector)

the orientation of a vector in space

the end point of a vector; the location of the tip of the vector’s arrowhead; also referred to as the “tip”

a method of adding vectors in which the tail of each vector is placed at the head of the previous vector

magnitude (of a vector)

the length or size of a vector; magnitude is a scalar quantity

resultant

the sum of two or more vectors

resultant vector

the vector sum of two or more vectors

scalar

a quantity with magnitude but no direction

tail

the start point of a vector; opposite to the head or tip of the arrow