2.9 Vector Addition

Definition

Vector addition is the process of combining two or more vectors.

If two vectors are represented by A→\vec{A} and B→\vec{B}, their vector sum is written as:

A→+B→\vec{A}+\vec{B}

Graphical Addition of Vectors

Head-to-Tail Method

To add two vectors graphically, place the tail of the second vector at the head of the first vector without changing its magnitude or direction.

The vector sum is represented by a vector drawn from the tail of the first vector to the head of the second vector.

Figure 2.9.1 – Vector addition using the head-to-tail method.

The order in which two vectors are added does not change their vector sum:

A→+B→=B→+A→\vec{A}+\vec{B}=\vec{B}+\vec{A}

This is called the commutative property of vector addition.

Parallelogram Method

Two vectors can also be added by placing their tails at the same point.

A parallelogram is then completed using the two vectors as adjacent sides. The diagonal drawn from the common tail represents their vector sum.

The head-to-tail method and the parallelogram method produce the same vector sum.

Adding More Than Two Vectors

More than two vectors can be added by placing them successively head to tail. For example:

A→+B→+C→\vec{A}+\vec{B}+\vec{C}

The vector sum is drawn from the tail of the first vector to the head of the final vector.

Figure 2.9.3 – Addition of three vectors using the head-to-tail method.

Vectors may be grouped in different ways without changing their vector sum:

A→+(B→+C→)=(A→+B→)+C→\vec{A}+\left(\vec{B}+\vec{C}\right)=\left(\vec{A}+\vec{B}\right)+\vec{C}

This is called the associative property of vector addition.

Adding Vectors Using Components

Vectors can be added by adding their corresponding components.

If

A→=Axi^+Ayj^\vec{A}=A_x\hat{i}+A_y\hat{j}

and

B→=Bxi^+Byj^\vec{B}=B_x\hat{i}+B_y\hat{j}

then

A→+B→=(Ax+Bx)i^+(Ay+By)j^\vec{A}+\vec{B}=(A_x+B_x)\hat{i}+(A_y+B_y)\hat{j}

Therefore, the x-components are added together:

(A→+B→)x=Ax+Bx(\vec{A}+\vec{B})_x=A_x+B_x

and the y-components are added together:

(A→+B→)y=Ay+By(\vec{A}+\vec{B})_y=A_y+B_y

Components along different axes cannot be added directly. An x-component must be added to another x-component, and a y-component must be added to another y-component.

Example

Two displacement vectors are given by:

A→=(3i^+2j^)m\vec{A}=(3\hat{i}+2\hat{j})\,\mathrm{m}

B→=(4i^−j^)m\vec{B}=(4\hat{i}-\hat{j})\,\mathrm{m}

Adding the corresponding components:

A→+B→=(3+4)i^+(2−1)j^\vec{A}+\vec{B}=(3+4)\hat{i}+(2-1)\hat{j}

A→+B→=(7i^+j^)m\vec{A}+\vec{B}=(7\hat{i}+\hat{j})\,\mathrm{m}

The vector sum has an x-component of 7 m and a y-component of 1 m.

Key Points

  • Vectors are added by considering both magnitude and direction.
  • In the head-to-tail method, the tail of each vector is placed at the head of the previous vector.
  • The parallelogram method is used when two vectors have a common tail.
  • The order of vector addition does not change the vector sum.
  • Only components along the same axis can be added.

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2.10 Vector Subtraction →