Addition and Subtraction of Vectors

Vectors are mathematical entities that have both magnitude and direction. They are used to represent quantities such as displacement, velocity, force, and many others in physics and engineering. Understanding how to add and subtract vectors is fundamental in these fields.

Vector Addition

Vector addition is the process of combining two or more vectors to form a single vector, known as the resultant vector. There are two main methods for adding vectors: the graphical method and the analytical method.

Graphical Method: The Triangle and Parallelogram Laws

Triangle Law of Addition

To add two vectors graphically using the triangle law, you place the tail of the second vector at the head of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the second vector.

Parallelogram Law of Addition

The parallelogram law involves placing two vectors such that they originate from the same point. A parallelogram is formed by drawing lines parallel to each vector through the head of the other. The resultant vector is drawn from the common origin to the opposite corner of the parallelogram.

Analytical Method: Component-Wise Addition

In the analytical method, vectors are broken down into their components, which are typically along the x, y, and possibly z axes in Cartesian coordinates. The components of the vectors are added algebraically.

Formula for Vector Addition

If $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$, then the resultant vector $\vec{R}$ is given by:

$$ \vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k} $$

Vector Subtraction

Vector subtraction is the process of finding a vector that represents the change from one vector to another. It can be thought of as adding a negative vector.

Graphical Method

To subtract vector $\vec{B}$ from vector $\vec{A}$ graphically, you reverse the direction of $\vec{B}$ to get $-\vec{B}$ and then add it to $\vec{A}$ using the triangle or parallelogram law.

Analytical Method: Component-Wise Subtraction

In the analytical method, you subtract the corresponding components of the vectors.

Formula for Vector Subtraction

If $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$, then the resultant vector $\vec{R}$ after subtracting $\vec{B}$ from $\vec{A}$ is given by:

$$ \vec{R} = \vec{A} - \vec{B} = (A_x - B_x)\hat{i} + (A_y - B_y)\hat{j} + (A_z - B_z)\hat{k} $$

Differences and Important Points

Here is a table summarizing the differences and important points of vector addition and subtraction:

Aspect	Vector Addition	Vector Subtraction
Definition	Combining two or more vectors to form a resultant vector.	Finding a vector that represents the change from one vector to another.
Graphical Method	Triangle Law or Parallelogram Law	Reverse the direction of the vector being subtracted and then add.
Analytical Method	Add corresponding components of the vectors.	Subtract corresponding components of the vectors.
Formula	$\vec{R} = \vec{A} + \vec{B}$	$\vec{R} = \vec{A} - \vec{B}$
Resultant Vector	Sum of the vectors	Difference between the vectors

Examples

Example 1: Graphical Addition of Vectors

Suppose we have two vectors $\vec{A}$ and $\vec{B}$ represented graphically. To find their sum $\vec{R}$ using the triangle law, we would draw vector $\vec{A}$, then place the tail of $\vec{B}$ at the head of $\vec{A}$. The resultant vector $\vec{R}$ would be drawn from the tail of $\vec{A}$ to the head of $\vec{B}$.

Example 2: Analytical Addition of Vectors

Let $\vec{A} = 3\hat{i} + 2\hat{j}$ and $\vec{B} = -1\hat{i} + 4\hat{j}$. To find $\vec{R}$, we add the components:

$$ \vec{R} = \vec{A} + \vec{B} = (3 - 1)\hat{i} + (2 + 4)\hat{j} = 2\hat{i} + 6\hat{j} $$

Example 3: Analytical Subtraction of Vectors

Let $\vec{A} = 5\hat{i} - 3\hat{j}$ and $\vec{B} = 2\hat{i} + 6\hat{j}$. To find $\vec{R}$, we subtract the components:

$$ \vec{R} = \vec{A} - \vec{B} = (5 - 2)\hat{i} - (3 + 6)\hat{j} = 3\hat{i} - 9\hat{j} $$

Understanding vector addition and subtraction is crucial for solving problems in physics and engineering where multiple vector quantities interact. The graphical method provides a visual understanding, while the analytical method allows for precise calculations.