Scalars, vectors and position vectors
A scalar has only magnitude; a vector has magnitude and direction. The position vector of P(x, y, z) is OP⃗ = xî + yĵ + zk̂.
Scalars and vectors
A scalar is a quantity fixed by one real number, its magnitude, such as 10 kg or 5 seconds. A vector needs a magnitude and a direction, such as 30 km/h towards the north. We draw a vector as a directed line segment AB⃗. A is the initial point and B is the terminal point. The length of the segment is the magnitude, written |AB⃗| or |a⃗|. It is never negative. The arrow from A to B shows the direction.
Scalar or vector?
| Quantity | Type | Why |
|---|---|---|
| Mass of a bag of rice, 10 kg | Scalar | No direction |
| Velocity of a train, 80 km/h towards Delhi | Vector | Speed with a direction |
| Temperature in Chennai, 34 °C | Scalar | No direction |
| Force of 20 N pulling a cart east | Vector | Size and direction |
| Distance walked, 5 km | Scalar | Path length only |
| Displacement, 3 km north-east of home | Vector | Straight-line change with a direction |
| Work done, 50 J | Scalar | Work has no direction, though force and displacement do |
Position vector of a point
Fix the origin O(0, 0, 0). For any point P(x, y, z), the vector OP⃗ is the position vector of P, often written r⃗. Using î, ĵ, k̂ (vectors of length 1 along the x, y and z axes), OP⃗ = xî + yĵ + zk̂. By Pythagoras' theorem used twice, its magnitude is |OP⃗| = √(x² + y² + z²), the distance of P from O.
AB⃗ vs BA⃗
|−a⃗| = |a⃗| always. Only the direction flips.
Example 1: position vector and its length
Find the position vector of P(2, −3, 6) and its magnitude.
- Position vector: OP⃗ = xî + yĵ + zk̂.
- OP⃗ = 2î − 3ĵ + 6k̂
- |OP⃗| = √(2² + (−3)² + 6²) = √(4 + 9 + 36)
- = √49 = 7
OP⃗ = 2î − 3ĵ + 6k̂ and |OP⃗| = 7 units.
Example 2: a drone flight
A drone takes off from a point O on the ground and reaches P(4, 4, 7), in metres. How far is it from O? Write the vector that brings it straight back to O.
- OP⃗ = 4î + 4ĵ + 7k̂
- |OP⃗| = √(16 + 16 + 49) = √81 = 9 m
- The return trip goes from P to O, so it is PO⃗ = −OP⃗.
- PO⃗ = −4î − 4ĵ − 7k̂, which also has magnitude 9 m.
The drone is 9 m from O. Return vector PO⃗ = −4î − 4ĵ − 7k̂.
A weather report saying 'wind at 20 km/h from the south-west' describes a vector. 'Temperature 34 °C' is a scalar, because it has no direction.
1 marks: “Classify as scalar or vector: (i) 10 J of work, (ii) a 5 N force acting downwards.” (i) Scalar: work has magnitude only. (ii) Vector: the force has a magnitude and a direction.
1 marks: “If |a⃗| = 4, what is |−a⃗|?” 4. The negative of a vector has the same magnitude and the opposite direction.
2 marks: “Find the distance of P(−2, 3, 6) from the origin and write the vector PO⃗.” OP⃗ = −2î + 3ĵ + 6k̂, |OP⃗| = √(4 + 9 + 36) = 7. PO⃗ = −OP⃗ = 2î − 3ĵ − 6k̂.
Scalar: how much. Vector: how much and which way.