How to Subtract Two Vectors Using Formula and Steps
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 Subtracting Two Vectors in Mathematics
1. What does subtracting two vectors mean?
Subtracting two vectors means adding the first vector to the negative (opposite) of the second vector. In vector algebra, A − B = A + (−B). The negative of a vector has the same magnitude but opposite direction. This operation gives a new vector that represents the difference in both magnitude and direction between the two vectors.
2. How do you subtract two vectors?
To subtract two vectors, subtract their corresponding components. If A = (a₁, a₂) and B = (b₁, b₂), then A − B = (a₁ − b₁, a₂ − b₂).
- Step 1: Write both vectors in component form.
- Step 2: Subtract each corresponding component.
- Step 3: Write the result as a new vector.
3. What is the formula for subtracting two vectors in component form?
The formula for subtracting two vectors in component form is (a₁, a₂, a₃) − (b₁, b₂, b₃) = (a₁ − b₁, a₂ − b₂, a₃ − b₃). Each component of the second vector is subtracted from the corresponding component of the first vector. This formula works for vectors in 2D, 3D, or higher dimensions.
4. Can you give an example of subtracting two vectors?
Yes, here is a simple example of vector subtraction. If A = (5, 7) and B = (2, 3), then:
- A − B = (5 − 2, 7 − 3)
- = (3, 4)
5. How do you subtract two 3D vectors?
To subtract two 3D vectors, subtract each of the x, y, and z components separately. If A = (x₁, y₁, z₁) and B = (x₂, y₂, z₂), then A − B = (x₁ − x₂, y₁ − y₂, z₁ − z₂).
- Subtract x-components.
- Subtract y-components.
- Subtract z-components.
6. What is the geometric interpretation of subtracting two vectors?
Geometrically, subtracting two vectors gives the vector from the tip of the second vector to the tip of the first vector when both start at the same point. In other words, A − B represents the direction and distance you move from B to reach A. This is often visualized using the head-to-tail method or the parallelogram rule.
7. Is subtracting vectors the same as adding negative vectors?
Yes, subtracting vectors is exactly the same as adding the negative of a vector. By definition, A − B = A + (−B). The vector −B has the same magnitude as B but points in the opposite direction, so vector subtraction follows the same rules as vector addition.
8. What is the difference between vector addition and vector subtraction?
The difference between vector addition and subtraction is that addition combines corresponding components, while subtraction subtracts them. For vectors A = (a₁, a₂) and B = (b₁, b₂):
- A + B = (a₁ + b₁, a₂ + b₂)
- A − B = (a₁ − b₁, a₂ − b₂)
9. Can you subtract vectors with different dimensions?
No, you cannot subtract vectors with different dimensions because corresponding components must match. For example, a 2D vector (a₁, a₂) cannot be subtracted from a 3D vector (b₁, b₂, b₃). Vector subtraction is only defined when both vectors have the same number of components.
10. What are common mistakes when subtracting two vectors?
A common mistake when subtracting two vectors is forgetting to subtract each corresponding component correctly. Key errors include:
- Changing the order of subtraction (remember A − B ≠ B − A).
- Forgetting to distribute the negative sign to all components.
- Mixing up components in 3D vectors.
