Vectors and Scalars
High-Yield Summary
- Vectors have magnitude + direction (displacement, velocity, acceleration, force); scalars have magnitude only (distance, speed, mass, energy).
- Vectors add graphically (tip-to-tail) or analytically (sum x- and y-components, then Pythagorean theorem + inverse tangent for the resultant).
- Vector subtraction is addition of the negative: A − B = A + (−B).
- Multiplying a vector by a negative scalar reverses its direction; the magnitude scales by |n|.
- Dot product (A·B) gives a scalar, used for work; cross product (A×B), found via the right-hand rule, gives a vector, used for torque/angular momentum.
Vector vs. Scalar Quantities
| Vector (magnitude + direction) | Scalar (magnitude only) |
|---|---|
| Displacement | Distance |
| Velocity | Speed |
| Acceleration | Mass |
| Force / weight | Energy, density, temperature |
Resultant Vector (Component Method)
R = √(Rx² + Ry²), θ = tan⁻¹(Ry / Rx)
- Rx, Ry = Sum of the x-components and y-components of the vectors being added
- R = Magnitude of the resultant vector
- θ = Direction of the resultant, measured from the x-axis
Dot Product vs. Cross Product
| Dot Product (A · B) | Cross Product (A × B) |
|---|---|
| Result: scalar | Result: vector |
| = |A||B| cos θ | = |A||B| sin θ n̂ |
| Maximum when parallel (θ = 0°) | Maximum when perpendicular (θ = 90°) |
| Zero when perpendicular (θ = 90°) | Zero when parallel (θ = 0° or 180°) |
| Used for: work | Used for: torque, angular momentum, magnetic fields |
Common MCAT Trap
- Cross product order matters: A × B and B × A point in opposite directions. Dot product order does not matter — same scalar either way.
- n < 0 scalar multiplication flips vector direction, not just scale — don't treat scalar sign as only affecting magnitude.
Quick Recall
What distinguishes a vector from a scalar?
When is the dot product zero, and when is the cross product zero?
How do you find the direction of a cross product's resultant vector?