Step-by-Step Method for Finding the Difference Between Vectors
When preparing for school or competitive exams, understanding subtracting two vectors helps you solve many real-life and exam problems where both magnitude and direction matter. Knowing this concept leads to easier calculations in physics, engineering, and coordinate geometry – all crucial for clear thinking in maths.
Formula Used in Subtracting Two Vectors
The standard formula is: \( \vec{A} - \vec{B} = \vec{A} + ( -\vec{B} ) \), where \( -\vec{B} \) is the same magnitude as \( \vec{B} \) but points in the opposite direction.
Here’s a helpful table to understand subtracting two vectors more clearly:
Subtracting Two Vectors Table
| Vector Operation | Formula | Resultant Direction |
|---|---|---|
| Addition (\( \vec{A} + \vec{B} \)) | Add respective components | Follows combined direction |
| Subtraction (\( \vec{A} - \vec{B} \)) | Subtract components: (\( A_x - B_x, A_y - B_y, A_z - B_z \)) | From tail of \( \vec{B} \) to tail of \( \vec{A} \) |
| Negative Vector (\( -\vec{B} \)) | Reverse direction of \( \vec{B} \) | Opposite to \( \vec{B} \) |
This table shows how subtracting two vectors results in a new direction and how it differs from simple addition.
Worked Example – Solving a Problem
Let’s subtract two vectors, \( \vec{A} = \langle 4, -2, 3 \rangle \) and \( \vec{B} = \langle 1, -2, 5 \rangle \), step by step:
1. Write the vectors in component form:2. Subtract corresponding components:
3. Simplify each component:
Final Answer: \( \vec{A} - \vec{B} = (3,\, 0,\, -2) \)
If you want more on the whole topic, see vector algebra for related properties and subtraction of vectors for JEE Main.
Practice Problems
- Subtract \( \vec{B} = (1, 3, -2) \) from \( \vec{A} = (4, -1, 5) \ ). Write your answer as a vector.
- Two position vectors are \( \vec{A}=(7,0) \) and \( \vec{B}=(2,5) \). Find the displacement vector \( \vec{A} - \vec{B} \).
- If \( \vec{P} = (2,4) \) and \( \vec{Q} = (3,-1) \), find \( \vec{P} - \vec{Q} \) and the magnitude of the result.
- If \( \vec{U} = (-3,2,1) \) and \( \vec{V} = (-1,7,4) \), compute \( \vec{U} - \vec{V} \).
Common Mistakes to Avoid
- Swapping the vector order: Remember, \( \vec{A} - \vec{B} \ne \vec{B} - \vec{A} \).
- Forgetting to change the direction of the subtracted vector, especially in graphical solutions.
- Confusing scalar subtraction with subtracting two vectors, which involves direction.
- Dropping a negative sign when subtracting vector components.
Real-World Applications
Subtracting two vectors is essential in navigation (finding the difference between two positions), physics (calculating relative velocity), and engineering (designing forces and resultant motions). For more fundamentals and how vectors connect to real scenarios, check vector and scalar quantities.
We explored the idea of subtracting two vectors, steps to apply it, worked example problems, and saw why it matters for real-world questions. Practice regularly, and use Vedantu’s resources on vector operations and joining two points with vectors to master this important topic.
FAQs on How to Subtract Two Vectors: Formula and Example
1. What is the formula for subtracting two vectors?
Vector subtraction is performed by subtracting the corresponding components of the two vectors. If vector A = (a1, a2, a3) and vector B = (b1, b2, b3), then their difference is A – B = (a1 – b1, a2 – b2, a3 – b3).
2. How can a vector be subtracted from another vector?
To subtract one vector from another, reverse the direction of the vector being subtracted (find its negative), and then add it to the first vector using the tip-to-tail method. Algebraically, subtract corresponding components.
3. What is the rule for subtraction vectors?
The rule for vector subtraction states: for two vectors A and B, A – B is equal to A + (–B), where –B is the vector with the same magnitude as B but opposite direction.
4. What is the formula for the difference of two vectors?
The difference formula for two vectors A and B is:
A – B = (a1 – b1, a2 – b2, ..., an – bn) for n components.
5. How do you subtract two vectors graphically?
Graphically, to subtract two vectors:
1. Reverse the direction of the vector being subtracted.
2. Place its tail at the head of the first vector.
3. The resultant vector from the tail of the first to the head of the second is the difference.
6. How can we subtract two vectors using Python or NumPy?
Using Python and NumPy, you can subtract vectors by:
1. Representing vectors as arrays (e.g., import numpy as np; A = np.array([a1, a2, a3])).
2. Subtract like this: C = A - B, which returns the difference component-wise.
7. How do you subtract two vectors in C++?
To subtract two vectors in C++, create arrays or use the vector class. Subtract each corresponding element in a loop. For built-in arrays:result[i] = A[i] - B[i]; for all components i.
8. How can we subtract two vectors in MATLAB?
In MATLAB, represent each vector as a row or column matrix. Subtract using:C = A - B
This performs element-wise subtraction.
9. How do you subtract two vectors in R?
In R, represent vectors as numeric arrays, then subtract directly:C <- A - B
This returns a vector with each element being the difference of the corresponding elements.
10. What is the difference between vector addition and subtraction?
The difference is:
Vector addition combines two vectors to form a resultant vector.
Vector subtraction finds the vector difference by adding the negative of one vector to the other.
11. How do you subtract multiple vectors?
To subtract multiple vectors, subtract each subsequent vector from the result of the previous operation: Result = A - B - C - D.... Algebraically, subtract corresponding components in order.
12. How do you subtract two points to find a vector?
To find a vector from two points P(x1, y1) and Q(x2, y2), subtract their coordinates: Q - P = (x2 – x1, y2 – y1). This vector represents the direction and magnitude from P to Q.
