Vectors on a grid
A Cartesian vector is written using its components along the coordinate axes:
- 2D: v = [x, y] = xi + yj
- 3D: v = [x, y, z] = xi + yj + zk
i, j and k are the unit vectors i, j, k: length 1 along the x-, y- and z-axes.
Resolving a vector
Resolving means splitting a vector into components.
For magnitude |v| at angle θ (from the positive x-axis):
x = |v| cos θ, y = |v| sin θ
Example: 50 N at 37°
- x = 50 cos 37° ≈ 39.9 N
- y = 50 sin 37° ≈ 30.1 N
- F ≈ [39.9, 30.1]
Why components are powerful
Operations become simple arithmetic:
| Operation | In components |
|---|---|
| addition | [a, b] + [c, d] = [a + c, b + d] |
| subtraction | [a − c, b − d] |
| scalar multiple | k[a, b] = [ka, kb] |
| magnitude | √(a² + b²) |
| dot product | ac + bd |
Three-dimensional vectors
In three-dimensional space, the axes are usually drawn with x toward you, y to the right and z up (a right-handed system).
Example: From A(1, −2, 3) to B(4, 2, 1):
AB⃗ = [3, 4, −2] = 3i + 4j − 2k
Its magnitude is |AB⃗| = √(9 + 16 + 4) = √29 ≈ 5.39
Vectors from the origin
A vector from the origin to point P(x, y, z) has the same components as the point’s coordinates: OP⃗ = [x, y, z]. (This is the position vector.)
Equality
Two Cartesian vectors are equal if and only if all their components are equal.
[a, 3, c] = [2, b, −1] → a = 2, b = 3, c = −1.
Example: combining forces
Three forces act on an object:
- F₁ = [10, 0]
- F₂ = [−4, 6]
- F₃ = [2, −9]
Net force = [8, −3]; magnitude ≈ 8.54 N.
Worked examples
Example 1: Write [4, −1, 6] using i, j, k. 4i − j + 6k.
Example 2: Resolve 100 m at 225°. [100 cos 225°, 100 sin 225°] ≈ [−70.7, −70.7].
Example 3: (2i − 3j) + (−i + 5j) = i + 2j.
Common mistakes
- Wrong signs for components in quadrants II–IV.
- Mixing up the order of x, y, z.
- Forgetting the k component in 3D.
Try it yourself
- Resolve 30 m/s at 150°.
- Write −2i + 7k in component form.
- Vector from P(0, 1, 2) to Q(3, −1, 5)?
- Magnitude of 2i − 3j + 6k?
Answers: 1) ≈ [−25.98, 15] 2) [−2, 0, 7] 3) [3, −2, 3] 4) 7
Quick recap
- A Cartesian vector uses components along the axes.
- Resolving: x = |v| cos θ, y = |v| sin θ.
- Unit vectors i, j, k make 3D vectors easy to write and combine.