Vector Subtraction Calculator with Steps
Calculator
The subtraction of vector A and vector B is a vector C derived by placing the initial point of B on the terminal point of A and then joining the initial point of A to the terminal point of B.
Vector Subtraction Formula
If A = (a₁i + b₁j + c₁k) and B = (a₂i + b₂j + c₂k), then the difference of vectors is:
A − B = (a₁−a₂)i + (b₁−b₂)j + (c₁−c₂)k
Vector subtraction is performed by subtracting corresponding components of the second vector from the first vector.
Magnitude of a Vector Difference
Once the difference vector is found, its magnitude is calculated using:
|A − B| = √[(a₁−a₂)² + (b₁−b₂)² + (c₁−c₂)²]
The magnitude represents the length of the resultant vector.
Direction Angles Formula
The direction angles of a vector are calculated using:
cos α = x/|V|
cos β = y/|V|
cos γ = z/|V|
where |V| is the magnitude of the resultant vector.
Comparison Table
| Operation | Formula |
|---|---|
| Addition | A + B |
| Subtraction | A − B |
| Dot Product | A · B |
| Cross Product | A × B |
Solved Example
Subtract the vectors:
A = (15i + 8j + 5k)
B = (15i + 8j + 5k)
Using the vector subtraction formula:
A − B = (15-15)i + (8−8)j + (5−5)k
A − B = 0i + 0j + 0k
Therefore the difference vector is: (0)
Solved Example 2:
Question:
Find the difference of the vectors:
A = (5i + 8j + 6k)
B = (7i + 2j + 1k)
Step 1: Write the Vector Subtraction Formula
A − B = (a₁−a₂)i + (b₁−b₂)j + (c₁−c₂)k
Step 2: Substitute the Values
A − B = (5−7)i + (8−2)j + (6−1)k
Step 3: Subtract Corresponding Components
A − B = (-2)i + (6)j + (5)k
A − B = -2i + 6j + 5k
Step 4: Calculate the Magnitude of the Resultant Vector
|A − B| = √[(-2)² + 6² + 5²]
= √(4 + 36 + 25)
= √65
= 8.0623
Step 5: Find the Direction Angles
cos α = x / |A−B|
= -2 / 8.0623
α = cos⁻¹(-0.2481)
α = 104.36°
cos β = y / |A−B|
= 6 / 8.0623
β = cos⁻¹(0.7442)
β = 41.99°
cos γ = z / |A−B|
= 5 / 8.0623
γ = cos⁻¹(0.6202)
γ = 51.67°
Final Answer
Difference Vector: -2i + 6j + 5k
Magnitude: 8.0623
Direction Angles:
- α = 104.36°
- β = 41.99°
- γ = 51.67°
Frequently Asked Questions
What is vector subtraction?
Vector subtraction is the process of finding the difference between two vectors by subtracting their corresponding components.
What is the formula for vector subtraction?
A − B = (a₁−a₂)i + (b₁−b₂)j + (c₁−c₂)k.
Can vectors be subtracted in 3D?
Yes. Each component is subtracted separately.
Why do we add a negative vector?
Because A − B is mathematically equivalent to A + (−B).
Where is vector subtraction used?
Physics, engineering, computer graphics, navigation, and machine learning.