Short on time? This page is the recap. For explanations, arrow diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- A scalar has a magnitude (size) only: distance, speed, time, mass.
- A vector has a magnitude and a direction: position, displacement, velocity, acceleration.
- Draw a vector as an arrow. It points in the vector’s direction, and its length is in proportion to its magnitude. Write the scale on the diagram.
- Write v⃗ (arrow on top) for the full vector. Along one axis, use the component v_x instead: a signed number with no arrow.
- The sign of a component gives the direction; its absolute value gives the magnitude. Speed = |v_x|.
- Changing the axis flips every sign but never changes the physical answer.
Key relationships
| Relationship | Meaning | Use it to |
|---|---|---|
| magnitude = |component| | size ignores sign | compare speeds or sizes |
| opposite direction ↔ opposite sign | the sign convention | write each vector as a signed number |
| sum = d₁ + d₂ + d₃ + … (signed) | vector sum (resultant) | find where an object ends up |
| total distance = |d₁| + |d₂| + … | scalar sum | find total path length |
| A − B = A + (−B) | subtraction adds the opposite | find a change, such as x − x₀ |
| v⃗ = v⃗₀ + a⃗t → v_x = v_x0 + a_x t | vector form → component form | substitute signed values |
Assumptions behind the numbers
- Motion is along one straight line, so a sign fully describes direction.
- Objects are treated as points.
- Near Earth’s surface, free-fall acceleration has size g = 9.8 m/s² directed downward (air resistance ignored).
Mistakes to avoid
- No axis stated. A sign means nothing without one.
- Adding sizes for a vector sum. That gives distance, not displacement.
- “Negative means small.” −8 m/s is faster than +5 m/s.
- Equal speeds treated as equal velocities. Check the directions too.
- Wrong sign on one term. Give every vector its sign before substituting, including g.
- Arrows not to scale. A longer arrow must mean a bigger magnitude.
Quick self-check
- Which of these are vectors: speed, acceleration, distance, position? (Acceleration and position.)
- With +x east, a cart moves 7.0 m east, 3.0 m west, then 6.0 m west. Find the net displacement and the distance. (−2.0 m, so 2.0 m west; distance 16 m.)
- With +x east, v_x0 = −4.0 m/s and a_x = +2.0 m/s². What is v_x after 3.0 s? (+2.0 m/s: 2.0 m/s east.)
- Which is faster: −8.0 m/s or +5.0 m/s? (−8.0 m/s; its speed is 8.0 m/s.)
Next: practice questions.