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103 LearnersLast updated on September 9, 2025
Subtraction of Two Vectors
The mathematical operation of finding the difference between two vectors is known as the subtraction of vectors. It helps in determining the resultant vector, which is crucial in physics and engineering for understanding directions and magnitudes.
What is Subtraction of Two Vectors?
Subtracting vectors involves adding the additive inverse of the second vector to the first. It requires changing the direction of the vector being subtracted and then combining the components. There are three components of a vector:
Magnitude: Represents the size or length of the vector.
Direction: Denotes the direction in which the vector is pointing.
Components: The horizontal and vertical projections of the vector.
How to Subtract Two Vectors?
When subtracting vectors, follow these steps:
Reverse direction: Reverse the direction of the vector to be subtracted and then add it to the first vector.
Add components: Subtract the corresponding components of the vectors to get the resultant vector.
Simplifying result: After performing the subtraction, you will have a new vector with its own magnitude and direction.
Methods to Subtract Two Vectors
The following are the methods of vector subtraction:
Method 1: Geometric Method
For the geometric method, follow these steps:
Step 1: Draw the first vector using a suitable scale.
Step 2: Draw the inverse of the second vector starting from the tip of the first vector.
Step 3: Connect the tail of the first vector to the tip of the inverted second vector.
Example: Subtract vector B from vector A.
Step 1: Draw vector A.
Step 2: Draw vector -B starting from the tip of vector A.
Step 3: The resultant vector is drawn from the tail of A to the tip of -B.
Method 2: Component Method
When using the component method, perform the following steps:
Step 1: Write the components of both vectors.
Step 2: Subtract the components of the second vector from the first vector.
Example: Subtract vector (3, 4) from vector (5, 6).
Solution: Subtract the components: First vector: (5, 6) Second vector: (3, 4) Resultant vector: (5-3, 6-4) = (2, 2)
Properties of Subtraction of Two Vectors
In vector mathematics, subtraction has some characteristic properties:
- Subtraction is not commutative In vector subtraction, changing the order of the vectors changes the result, i.e., A - B ≠ B - A.
- Subtraction is the addition of the opposite
- Subtracting a vector is the same as adding its opposite, so subtraction can be converted into addition by changing the direction of the second vector.
- A - B = A + (-B) Zero vector Subtracting a vector from itself results in a zero vector. A - A = 0
Tips and Tricks for Subtraction of Two Vectors
Here are some tips for efficiently subtracting vectors:
Tip 1: Always pay attention to the direction when adding the inverse vector.
Tip 2: Use graph paper for geometric methods to maintain accuracy in drawing vectors.
Tip 3: Double-check component calculations to avoid errors in the resultant vector.
Ignoring direction change
Students often forget to reverse the direction of the vector being subtracted. Always remember to reverse the direction before performing addition.
Mistake 1
Mixing up components
Only subtract corresponding components; mixing them up leads to incorrect results.
Mistake 2
Inaccurate drawings in geometric method
Ensure vectors are drawn to scale and directions are accurately represented for correct results in the geometric method.
Mistake 3
Leaving vectors unsimplified
Always check for simplification of the resultant vector to ensure it is expressed in its simplest form.
Mistake 4
Overlooking zero vector outcome
Subtracting a vector from itself results in a zero vector. This is often overlooked but is an important concept.
Mistake 5
Examples of Subtraction of Two Vectors
Subtract vector (2, 3) from vector (5, 7)
(3, 4)
Problem 1
Use the component method: (5, 7) - (2, 3) = (5-2, 7-3) = (3, 4)
Subtract vector (4, 1) from vector (6, 5)
Explanation
(2, 4)
Problem 2
Use the component method: (6, 5) - (4, 1) = (6-4, 5-1) = (2, 4)
Subtract vector (1, -2) from vector (-3, 4)
Explanation
(-4, 6)
Problem 3
(-3, 4) - (1, -2) = (-3-1, 4+2) = (-4, 6)
Subtract vector (3, 6) from vector (8, 2)
Explanation
(5, -4)
Problem 4
(8, 2) - (3, 6) = (8-3, 2-6) = (5, -4)
Subtract vector (4, 5) from vector (9, 8)
Explanation
(5, 3)
No, vectors must have the same dimensions to be subtracted; otherwise, the operation is undefined.
1.Is vector subtraction commutative?
2.What is the zero vector?
3.How do we represent the subtraction of vectors geometrically?
4.What methods are used for the subtraction of vectors?
Common Mistakes and How to Avoid Them in Subtraction of Two Vectors
Subtraction of vectors can be challenging, and there are common mistakes students should be aware of to avoid them.
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
