Author Jonathan David
Author Jonathan David
Lessons

Vectors and Vector Addition

by |Published |1 comment
Vectors and Vector Addition

Vectors and Vector Addition

What Are Vectors?

A vector is a quantity that has both magnitude and direction. Vectors are used to represent physical quantities like displacement, velocity, force, and acceleration.

For example, a car traveling 50 \, \text{km/h} north can be represented by a vector with:

  • Magnitude: 50 \, \text{km/h}.
  • Direction: North.

Vector Representation

Vectors are typically represented by arrows:

  • The length of the arrow indicates the magnitude.
  • The direction of the arrow represents the vector’s direction.

Mathematically, a vector \vec{A} can be expressed in component form:

    \[\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}\]

Here:

  • A_x, A_y, A_z: Components of the vector along the x-, y-, and z-axes.
  • \hat{i}, \hat{j}, \hat{k}: Unit vectors along the x-, y-, and z-axes.

Vector Addition

Vectors can be added geometrically or algebraically:

Geometric Method (Graphical Addition)

To add two vectors \vec{A} and \vec{B}:

  1. Place the tail of \vec{B} at the head of \vec{A}.
  2. The resultant vector \vec{R} is drawn from the tail of \vec{A} to the head of \vec{B}.

This method is also called the head-to-tail method.

Algebraic Method (Component Addition)

When vectors are expressed in components:

    \[\vec{A} = A_x \hat{i} + A_y \hat{j}, \quad \vec{B} = B_x \hat{i} + B_y \hat{j}\]

The resultant vector \vec{R} = \vec{A} + \vec{B} is:

    \[\vec{R} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}\]

Example Problem

Add the vectors \vec{A} = 3 \hat{i} + 4 \hat{j} and \vec{B} = -2 \hat{i} + 5 \hat{j}.

Solution:

    \[\vec{R} = \vec{A} + \vec{B} = (3 + (-2)) \hat{i} + (4 + 5) \hat{j}\]

    \[\vec{R} = 1 \hat{i} + 9 \hat{j}\]

The resultant vector is \vec{R} = \hat{i} + 9 \hat{j}.

Applications of Vector Addition

Vector addition is used in:

  • Calculating net forces acting on an object.
  • Finding resultant velocities in relative motion problems.
  • Determining displacement in navigation and kinematics.

Key Takeaways

Vectors are essential for describing quantities with both magnitude and direction. Vector addition can be performed geometrically or algebraically, and it is a fundamental tool in physics for solving problems involving forces, motion, and more.