Scalars vs Vectors
In physics, we categorize quantities based on whether they include direction. This gives us two types of quantities:
Scalar Quantities
Scalars are quantities that have only magnitude (a numerical value) and no direction.
Examples: Distance, speed, mass, temperature, energy
Vector Quantities
Vectors have both magnitude and direction. They are often represented by arrows, where the length indicates magnitude and the arrow points in the direction.
Examples: Displacement, velocity, acceleration, force
Visual Example
Consider a person walking 5 meters east. That's a vector quantity. But if you only care about how far they walked, regardless of direction, that's a scalar.
Vector Notation
In physics, vectors can be written in multiple ways:
- Arrow notation: \( \vec{v} \)
- Boldface: v
- Component form: \( \vec{v} = \langle 3, 4 \rangle \)
- Unit vector form: \( \vec{v} = 3\hat{\imath} + 4\hat{\jmath} \) (where \( \hat{\imath} \) is the unit vector in the +x direction, \( \hat{\jmath} \) is the unit vector in the +y direction, and \( \hat{\mathbf{k}} \) is the unit vector in the +z direction)
Example: \( \vec{v} = 5\hat{\imath} - 2\hat{\jmath} + 7\hat{k} \)
Vector Operations
Vectors can be added, subtracted, and multiplied by scalars. For example:
- Addition: \( \vec{a} + \vec{b} = \langle a_x + b_x, a_y + b_y \rangle \)
- Subtraction: \( \vec{a} - \vec{b} = \langle a_x - b_x, a_y - b_y \rangle \)
- Scalar multiplication: \( 2\vec{v} = \langle 2v_x, 2v_y \rangle \)
Applications in Physics
Vectors are everywhere in physics. Here are a few examples:
- Displacement: Moving from one point to another
- Velocity: Speed with direction
- Acceleration: Changing velocity over time
- Force (next unit): Combines magnitude and direction to determine motion
Key Takeaways
- Vectors have both magnitude and direction
- They can be represented using arrows, components, or unit vectors
- They're essential in understanding real-world motion and forces