Vectors
Magnitude, the dot product and perpendicular vectors.
Learn with Kutty
Go one screen at a time. Press Next when you are ready. Tap 🔊 Read to me to listen.
A vector is an arrow: it has a size and a direction.
Look: Follow a, then b. The green arrow is a + b.
✏️ Your turn!
Type your answer and press Check. Kutty will tell you if it is right.
1Length of 6i + 8j?
2(1, 2) + (3, 4) = ?
3Length of 5i − 12j?
! Careful!
✗Adding the lengths: 3 + 4 = 7.
✓Different directions: use Pythagoras (5).
✗Thinking direction does not matter.
✓Same size, different direction: different vector.
See the whole lesson on one page
A vector is an arrow: it has a size and a direction.
Add vectors tip to tail: follow one arrow, then the next. The arrow from start to finish is the sum. Its length comes from Pythagoras.
Think of it like this"Walk 3 steps east, then 4 steps north" — each instruction is a vector. Together they take you to a new spot.
- Magnitude
- The length of the arrow.
Solved together
Walk 3 east and 4 north. How far are you from the start?
The two walks are at right angles.
3i + 4jPythagoras for the straight-line distance.
√(3² + 4²) = √25 = 5
Answer5 steps away.
For older learners and grown-ups: a little more
Dot product a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ; it is 0 exactly when the vectors are perpendicular. Example: (2, 1, −2)·(1, 2, 2) = 0.
Now practise it
Kanakku makes new questions on vectors every time. It shows the working and gives a hint when you are stuck.