102 LearnersLast updated on August 30, 2025
Subtraction of Two Vectors
The mathematical operation of finding the difference between two vectors is known as the subtraction of vectors. It helps simplify vector problems and solve tasks involving magnitudes, directions, and vector operations.
What is Subtraction of Two Vectors?
Subtracting vectors involves adding the additive inverse of the second vector to the first.
It requires reversing the direction of the vector being subtracted and then combining the vectors.
There are three components of a vector:
Magnitude: This is the length or size of the vector.
Direction: This indicates the direction in which the vector acts.
Notation: For subtraction, vectors are typically written using notation like v - w.
How to do Subtraction of Two Vectors?
When subtracting vectors, students should follow these steps:
Reverse direction: Reverse the direction of the second vector and perform addition of vectors.
Combine vectors: Add the reversed vector to the first vector using vector addition rules.
Simplifying result: After addition, simplify the result to obtain the final vector.
Methods to do Subtraction of Two Vectors
The following are methods of subtracting vectors:
Method 1: Graphical Method
To apply the graphical method for vector subtraction, use the following steps.
Step 1: Draw the first vector using a suitable scale and direction.
Step 2: Draw the second vector in the opposite direction using the same scale.
Step 3: Connect the initial point of the first vector to the endpoint of the reversed second vector to find the resultant vector.
Example: Subtract vector w from vector v.
Step 1: Draw v.
Step 2: Reverse and draw w.
Step 3: Draw the resultant vector from the start of v to the end of the reversed w.
Method 2: Analytical Method
In the analytical method, we use vector components.
Subtract the corresponding components of the vectors to find the resultant vector.
For example, Subtract w(2, −3) from v(5, 4). v(5, 4) - w(2, −3) = (5−2, 4−(−3)) = (3, 7)
Therefore, the resultant vector is (3, 7).
Properties of Subtraction of Two Vectors
In vector math, subtraction has characteristic properties.
These properties are listed below:
Subtraction is not commutative In vector subtraction, changing the order changes the result, i.e., v - w ≠ w - v.
Subtraction is not associative Unlike addition, we cannot regroup in subtraction.
When three or more vectors are involved, changing the grouping changes the result.
(v − w) − u ≠ v − (w − u) Subtraction is the addition of the opposite Subtracting a vector is the same as adding its opposite, so to simplify calculations, you can convert subtraction into addition by reversing the second vector.
v − w = v + (−w) Subtracting zero vector leaves the vector unchanged Subtracting the zero vector from any vector results in the same vector: v - 0 = v.
Tips and Tricks for Subtraction of Two Vectors
Tips and tricks are useful for students to efficiently handle vector subtraction.
Some helpful tips are listed below:
Tip 1: Always pay attention to vector directions before combining them.
Tip 2: If two vectors have identical directions and magnitudes, they cancel each other out when subtracted.
Tip 3: Beginners can use vector diagrams to visualize subtraction and avoid errors.
Forgetting to reverse the direction
Students often forget to reverse the direction of the vector being subtracted.Always reverse before combining.
Mistake 1
Misalignment in vector components
When using components, ensure the correct components are subtracted.Mixing them leads to incorrect results.
Mistake 2
Ignoring vector magnitude and direction
Remember to consider both magnitude and direction when subtracting.Ignoring either can lead to errors.
Mistake 3
Leaving vectors unsimplified
Always check for possible simplifications after subtraction to ensure the vector is in its simplest form.
Mistake 4
Misinterpreting vector notation
Misunderstanding vector notation can lead to errors.Ensure you understand the notation used for vector subtraction.
Mistake 5
Examples of Subtraction of Two Vectors
Subtract vector b(3, 1) from vector a(5, 4)
(2, 3)
Problem 1
Use the analytical method, a(5, 4) - b(3, 1) = (5−3, 4−1) = (2, 3)
Subtract v(7, 2) from u(10, 4)
Explanation
(3, 2)
Problem 2
Using the analytical method of subtraction u(10, 4) - v(7, 2) = (10−7, 4−2) = (3, 2)
Subtract vector p(−2, 5) from vector q(4, 3)
Explanation
(6, -2)
Problem 3
q(4, 3) - p(−2, 5) = (4−(−2), 3−5) = (6, -2)
Subtract vector r(3, -4) from vector s(6, 1)
Explanation
(3, 5)
Problem 4
s(6, 1) - r(3, -4) = (6−3, 1−(−4)) = (3, 5)
Subtract c(1, 2) from d(3, 5)
Explanation
(2, 3)
No, vectors must have the same dimensions to be subtracted; otherwise, subtraction is undefined.
1.Is subtraction commutative for vectors?
2.What is the zero vector?
3.What is the first step of vector subtraction?
4.What methods are used for vector subtraction?
5.How can children in Thailand use numbers in everyday life to understand Subtraction of Two Vectors?
6.What are some fun ways kids in Thailand can practice Subtraction of Two Vectors with numbers?
7.What role do numbers and Subtraction of Two Vectors play in helping children in Thailand develop problem-solving skills?
8.How can families in Thailand create number-rich environments to improve Subtraction of Two Vectors skills?
Common Mistakes and How to Avoid Them in Subtraction of Two Vectors
Vector subtraction can be challenging, leading to common mistakes.However, being aware of these errors can help students avoid them.
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.
