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102 LearnersLast updated on September 8, 2025
Subtraction of Vectors
The mathematical operation of finding the difference between two vectors is known as the subtraction of vectors. It helps in understanding relative positions, directions, and magnitudes in physics and engineering problems.
What is Subtraction of Vectors?
Subtracting vectors involves adding the additive inverse of the second vector to the first. This involves reversing the direction of the second vector and then performing vector addition.
A vector has three components:
Magnitude: The length or size of the vector.
Direction: The orientation of the vector in space.
Components: The projections of the vector along the coordinate axes.
How to do Subtraction of Vectors?
When subtracting vectors, follow these rules:
Reverse direction: Reverse the direction of the vector being subtracted by changing its sign.
Add vectors: Perform vector addition by adding the corresponding components of the vectors.
Simplifying result: The result is a new vector representing the difference between the original vectors.
Methods to do Subtraction of Vectors
The following are methods of subtracting vectors:
Method 1: Graphical Method
To apply the graphical method for vector subtraction, use these steps:
Step 1: Draw the first vector using appropriate scale and direction.
Step 2: Draw the second vector from the head of the first vector but in the opposite direction.
Step 3: The resultant vector from the tail of the first vector to the head of the second vector is the difference.
Example: Subtract vector B from vector A.
Step 1: Draw vector A.
Step 2: Draw vector B starting from the head of A but in the opposite direction.
Step 3: The resultant vector from the start of A to the end of the reversed B is A - B.
Method 2: Analytical Method
Subtract vectors using their components. Write the vectors in component form and subtract the corresponding components.
Example: Subtract B = <2, -3> from A = <5, 4>.
Solution: A - B = <5 - 2, 4 - (-3)> = <3, 7>
Therefore, the resultant vector is <3, 7>.
Properties of Subtraction of Vectors
Vector subtraction has certain properties:
- Subtraction is not commutative Changing the order of vectors changes the result, i.e., A - B ≠ B - A.
- Subtraction is not associative When three or more vectors are involved, changing the grouping changes the result. (A − B) − C ≠ A − (B − C)
- Subtraction as addition of the negative Subtracting a vector is the same as adding its negative, converting subtraction into addition by reversing the vector's direction. A − B = A + (−B)
Tips and Tricks for Subtraction of Vectors
The following tips are helpful for vector subtraction:
Tip 1: Always pay attention to the direction before subtracting vectors.
Tip 2: For ease, convert vectors into component form and handle subtraction as component-wise operations.
Tip 3: Use vector diagrams for visual learners to better understand direction changes and resultant vectors.
Forgetting to reverse direction
Students often forget to reverse the direction of the vector being subtracted. Always remember to reverse the direction of the vector before adding.
Mistake 1
Incorrect component subtraction
Only corresponding components should be subtracted; mixing components leads to incorrect results.
Mistake 2
Misalignment in graphical representation
When using the graphical method, ensure the direction and scale of vectors are accurately represented.
Mistake 3
Leaving vectors without simplification
Always check for component simplification in the final result.
Mistake 4
Ignoring zero vectors
If a vector is missing a component, consider it as zero. For example, if the z-component is missing, it can be written as 0.
Mistake 5
Examples of Subtraction of Vectors
Subtract vector B = <2, 1> from vector A = <6, 3>
<4, 2>
Problem 1
Use the component method: A - B = <6 - 2, 3 - 1> = <4, 2>
Subtract vector D = <4, -5> from vector C = <7, 2>
Explanation
<3, 7>
Problem 2
Use the component method: C - D = <7 - 4, 2 - (-5)> = <3, 7>
Subtract vector F = <-3, 2> from vector E = <1, -4>
Explanation
<4, -6>
Problem 3
E - F = <1 - (-3), -4 - 2> = <4, -6>
Subtract vector H = <0, 3> from vector G = <5, 5>
Explanation
<5, 2>
Problem 4
G - H = <5 - 0, 5 - 3> = <5, 2>
Subtract vector J = <2, -3, 1> from vector I = <4, 1, -2>
Explanation
<2, 4, -3>
No, vectors must have the same dimensions to be subtracted; they must have the same number of components.
1.Is vector subtraction commutative?
2.What are the vector components?
3.What is the first step in vector subtraction?
4.What methods are used for vector subtraction?
Common Mistakes and How to Avoid Them in Subtraction of Vectors
Subtracting vectors can be tricky, leading to common mistakes. Awareness of these errors can help 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.
