Difference between tensor and vector

 

1. VECTOR (Rank-1 Tensor)

A vector is a quantity that has both magnitude and direction in space.

✅ Key Properties:

One-dimensional array of numbers.

Rank = 1

Examples:

Velocity: 50 km/h east

Force: 10 N upward
Displacement: 5 m north

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🔸 2. TENSOR (Generalized Concept)

A tensor is a mathematical object that generalizes scalars, vectors, and matrices to higher dimensions.

✅ Key Properties:

Can have any rank: 0 (scalar), 1 (vector), 2 (matrix), etc.

Represented as multi-dimensional arrays.
Follows specific transformation rules under a change of coordinates.

Tensor Type	Rank	Example
Scalar	0	Temperature, Mass
Vector	1	Velocity, Force
Matrix (2D Tensor)	2	Stress tensor, Moment of inertia
Higher-order	3+	Used in elasticity, relativity

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🔍 Main Differences at a Glance

Feature	Vector	Tensor
Rank	1	Can be 0, 1, 2, 3, ...
Dimensions	One-dimensional array	Multi-dimensional array
Example	Velocity, Force	Stress, Strain, Inertia Tensor
Transformation rule	Linear under rotation	More complex (depending on rank)
Represented as	List of numbers	Matrix or higher-dimensional array

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💡 Example:

1. Vector (1st-order tensor)

2. Tensor (2nd-order tensor):

$$\text{}$$

This matrix tells how internal forces (stresses) act inside a solid in all directions.

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✅ Summary in Simple Words:

Every vector is a tensor, but not every tensor is a vector.
Tensors are a broader, more powerful concept used in physics (like relativity), engineering (stress/strain), and machine learning.