Vectors Calculator
Calculate vector operations with step-by-step solutions. Supports 2D and 3D vectors for GCSE and A-Level.
Step-by-step learning with explanations
Vectors Calculator
Enter Vector Details
Vector a
2DVector b
2DEnter vector values and click Calculate to view the result and geometric interpretation.
To add vectors, you add their corresponding components.
Remember that . You must add the vectors first, then find the magnitude.
Example
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Understanding Vectors
What is a Vector?
A vector is a quantity with both magnitude (size) and direction. Written as column vectors or coordinate pairs like (3, 4) or (1, 2, 5).
Examples:
Displacement, velocity, force, acceleration
Vectors vs Scalars
- Vectors:Have magnitude AND direction (e.g., velocity)
- Scalars:Have magnitude only (e.g., speed, temperature)
- •The dot product of vectors gives a scalar
- •The cross product of vectors gives a vector
Key Vector Operations
Addition & Subtraction
Add or subtract corresponding components.
(3, 4) + (1, 2) = (4, 6)
3+1=4, 4+2=6
(5, 8) - (2, 3) = (3, 5)
5-2=3, 8-3=5
Dot Product (Scalar Product)
Multiply corresponding components and add. Result is a scalar.
a · b = a₁b₁ + a₂b₂ + a₃b₃
(3, 4) · (1, 2) = 3×1 + 4×2 = 3 + 8 = 11
💡 If a · b = 0, vectors are perpendicular!
Cross Product (3D Only)
Produces a vector perpendicular to both inputs. Order matters!
a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
(1, 0, 0) × (0, 1, 0) = (0, 0, 1)
⚠️ a × b ≠ b × a (not commutative!)
Magnitude (Length)
Use Pythagoras in multiple dimensions.
2D:
|v| = √(x² + y²)
3D:
|v| = √(x² + y² + z²)
Exam Tips
Perpendicular Check
If a · b = 0, vectors are perpendicular. This is a common exam question!
Parallel Check
Vectors are parallel if one is a scalar multiple of the other: a = kb.
Unit Vectors
Always simplify to unit vectors when asked. Divide by magnitude.
Cross Product Order
Remember: a × b = -(b × a). Order matters for cross products!
Position Vectors
AB = b - a (end minus start). Common mistake is subtracting wrong way.
Exact Form
Leave answers as surds (√5) unless told to give decimals.
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