What is a Vector?

A vector is a physical quantity that has both magnitude and direction. Examples of vector quantities include displacement, velocity, acceleration, and force.

Scalar quantities, on the other hand, have only magnitude and no direction, e.g., mass, temperature, time, energy.

Representation of Vectors

Vectors are represented graphically by arrows:

  • Length of the arrow indicates the magnitude.
  • Direction of the arrow indicates the direction of the vector.

Notation:

  • A vector is usually denoted as A⃗\vec{A} or A.

  • Magnitude of vector A⃗\vec{A} is denoted as ∣A⃗∣|\vec{A}| or just AA.

    Speed vs Velocity Comparison

    This diagram shows a vector A⃗\vec{A} making an angle θ\theta with the horizontal.

Types of Vectors

1. Zero or Null Vector (0⃗\vec{0}):

  • A vector with zero magnitude and undefined direction.
  • Example: Displacement of a stationary object.

2. Unit Vector:

  • A vector of unit magnitude.
  • Denotes only direction.
  • Represented as: A^=A⃗∣A⃗∣\hat{A} = \frac{\vec{A}}{|\vec{A}|}

3. Equal Vectors:

  • Vectors having the same magnitude and direction.
  • May originate from different points.

4. Negative Vectors:

  • Same magnitude but opposite direction.
  • A⃗\vec{A} and −A⃗-\vec{A} are negative of each other.

5. Collinear or Parallel Vectors:

  • Vectors lying along the same or parallel lines.
  • Can be in same or opposite direction.

6. Coplanar Vectors:

  • Vectors lying in the same plane.

7. Position Vector:

  • A vector that locates a point with respect to origin.
  • Example: r⃗=xi^+yj^+zk^\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}.

8. Co-initial Vectors:

  • Vectors having same initial point.

9. Concurrent Vectors:

  • Vectors acting on a common point.

10. Orthogonal Vectors:

  • Vectors at 90° to each other.

Example:

Question: A car moves 5 km east and then 12 km north. Find the displacement vector and its magnitude.

Solution: Let east be along x-axis and north along y-axis. The displacement vector is: R⃗=5i^+12j^\vec{R} = 5\hat{i} + 12\hat{j} Magnitude of R⃗\vec{R} is: ∣R⃗∣=52+122=25+144=169=13 km|\vec{R}| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\ km

Example:

Question: A vector A⃗\vec{A} has magnitude 10 units and another vector B⃗=−A⃗\vec{B} = -\vec{A}. What is the relation between them?

Answer: Both vectors have same magnitude (10), but opposite directions. Hence, they are negative vectors of each other.