Vectors in Physics

Study Notes

Definition

A vector is a quantity that has both magnitude and direction.
Common examples include displacement, velocity, acceleration, and force.

Representation

Vectors are often represented graphically as arrows:
- The length of the arrow indicates the magnitude.
- The direction of the arrow indicates the direction of the vector.
Algebraically, a vector can be expressed in component form: v = (v₁, v₂) for 2D or v = (v₁, v₂, v₃) for 3D.

Operations

Addition
- Vectors can be added graphically using the tip-to-tail method or mathematically by adding corresponding components.
- If A = (a₁, a₂) and B = (b₁, b₂), then A + B = (a₁ + b₁, a₂ + b₂).
Subtraction
- To subtract vectors, reverse the direction of the vector being subtracted.
- A - B = A + (-B).
Scalar Multiplication
- A vector can be multiplied by a scalar, changing its magnitude but not its direction (if the scalar is positive).
- If k is a scalar and A = (a₁, a₂), then kA = (ka₁, ka₂).

Magnitude

The magnitude of a vector v = (v₁, v₂) is calculated as:
- |v| = √(v₁² + v₂²) in 2D.
- |v| = √(v₁² + v₂² + v₃²) in 3D.

Unit Vectors

A unit vector is a vector with a magnitude of 1.
It indicates direction only and can be found by dividing a vector by its magnitude.
Common unit vectors in 2D:
- i = (1, 0)
- j = (0, 1)

Dot Product

The dot product of two vectors A and B is a scalar and is calculated as:
- A · B = a₁b₁ + a₂b₂ for 2D vectors.
The dot product is related to the angle θ between vectors by:
- A · B = |A||B|cos(θ).

Cross Product

The cross product is defined only in three dimensions and yields a vector.
If A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), then the cross product is:
- A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).
The resultant vector is perpendicular to both A and B.

Applications

Used in physics to describe forces, motion, and fields.
Essential in engineering, computer graphics, and navigation.

Vectors

A vector is a quantity that has both magnitude and direction.
Examples include displacement, velocity, acceleration, and force.
Represented graphically as arrows; the length indicates magnitude and the direction the arrow points is the direction of the vector.
Can be expressed in component form; **v** = (v₁, v₂) for 2D or **v** = (v₁, v₂, v₃) for 3D.

Vector Operations

Addition can be done graphically using the tip-to-tail method or mathematically by adding corresponding components.
If **A** = (a₁, a₂) and **B** = (b₁, b₂) then **A** + **B** = (a₁ + b₁, a₂ + b₂)
Subtraction involves reversing the direction of the vector being subtracted. `A - B = A + (-B).
Scalar Multiplication involves multiplying a vector by a scalar, which changes its magnitude but not its direction (if the scalar is positive). k**A** = (ka₁, ka₂).

Vector Magnitude

The magnitude of a vector is calculated by using the Pythagorean theorem:
|**v**| = √(v₁² + v₂²) in 2D
|**v**| = √(v₁² + v₂² + v₃²) in 3D.

Unit Vectors

Unit vectors have a magnitude of 1, indicating direction only, and are found by dividing a vector by its magnitude.
Common unit vectors in 2D:
- i = (1, 0)
- j = (0, 1)

Dot Product

The dot product is a scalar.
Calculated as: **A** · **B** = a₁b₁ + a₂b₂ for 2D vectors.
Measures the "projection" of one vector onto another.
**A** · **B** = |**A**||**B**|cos(θ).

Cross Product

Defined only in three dimensions and yields a vector.
If **A** = (a₁, a₂, a₃) and **B** = (b₁, b₂, b₃) then the cross product:
**A** × **B** = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁).
The resultant vector is perpendicular to both A and B.

Vector Applications

Used in physics to describe forces, motion, and fields.
Essential in engineering, computer graphics, and navigation.

Description

This quiz covers the fundamental concepts of vectors, including their definition, representation, and operations like addition, subtraction, and scalar multiplication. Test your understanding of vectors and their applications in physics with this engaging quiz.