Short on time? This page is the recap for the calculus-based course, which uses î, ĵ, k̂ notation in three dimensions (the Physics 1 topic stays in one dimension). For explanations, diagrams and worked examples, use the full study guide.
Recap
- Scalars have magnitude only: distance, speed, time, mass.
- Vectors have magnitude and direction: position, displacement, velocity, acceleration.
- Draw a vector as an arrow: it points the vector’s way, and its length is proportional to the magnitude (state the scale).
- In one dimension, state the positive direction; opposite directions get opposite signs.
- In two or three dimensions, write vectors with unit vectors î, ĵ, k̂ along +x, +y, +z, or as a magnitude and a direction.
- The resultant is found by adding components axis by axis.
Key relationships
| Relationship | Meaning |
|---|---|
| A = A_x î + A_y ĵ + A_z k̂ | unit vector notation; components are signed scalars |
| |A| = √(A_x² + A_y² + A_z²) | magnitude, never negative |
| A_x = |A| cos θ, A_y = |A| sin θ | θ measured from +x towards +y |
| tan θ = A_y / A_x | direction; check the quadrant with a sketch |
| r = x î + y ĵ + z k̂ | position vector from the origin to the point |
| r̂ = r / |r| | unit vector along r: magnitude 1, no unit |
| Δr = r − r₀ | displacement |
| R_x = A_x + B_x, R_y = A_y + B_y, R_z = A_z + B_z | resultant R = A + B |
| |A| − |B| ≤ |A + B| ≤ |A| + |B| | range of a resultant (|A| ≥ |B|) |
Assumptions behind the numbers
- Axes are perpendicular and right-handed; î, ĵ, k̂ each have magnitude 1.
- Angles are in degrees unless stated. Your calculator must be in degree mode.
- Vectors may be moved on a diagram if their length and direction are unchanged.
Mistakes to avoid
- Adding magnitudes instead of components.
- Distance = |displacement| for a path that bends or turns back.
- Trusting tan⁻¹ without checking which quadrant the vector is in.
- Swapping sine and cosine when the angle is measured from the y-axis.
- Dropping a sign for a component that points west, south, down or left.
- Giving r̂ a unit, or a magnitude other than 1.
- |r₂| − |r₁| for |Δr|. Subtract the vectors first.
Quick self-check
- What is the magnitude of (2.0 î − 6.0 ĵ + 3.0 k̂) m? (√(4 + 36 + 9) = 7.0 m)
- A displacement is 20 m at 30° below the +x axis. Give its components. (17.3 m and −10.0 m)
- Add (4.0 î + 1.0 ĵ) m and (−6.0 î + 3.0 ĵ) m. Give the magnitude and the angle from +x. ((−2.0 î + 4.0 ĵ) m; 4.47 m at 116.6°, second quadrant)
- With +x to the right, a cart moves 2.5 m/s to the left. What is v_x? (−2.5 m/s; speed 2.5 m/s)
Next: practice questions.