Two vectors are equal if and only if they have the same magnitude and direction.


To understand why the correct answer is option 'D', let's break down the components of a vector and what it means for two vectors to be equal.


A vector is a mathematical object that has both magnitude and direction. Vectors are typically represented by an arrow, where the length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.


There are two main components of a vector: magnitude and direction.

1. Magnitude: The magnitude of a vector refers to its size or length. It is always a positive value and represents the distance or quantity associated with the vector. Magnitude is a scalar quantity and can be represented by a positive number.

2. Direction: The direction of a vector refers to the orientation or angle of the vector with respect to a reference. It is represented by an arrow pointing in a particular direction.


For two vectors to be considered equal, they must possess the same magnitude and direction. Let's consider the different possibilities to understand why this is the case:

1. Same Magnitude but Different Directions: If two vectors have the same magnitude but different directions, they cannot be considered equal. This is because the direction of a vector is an essential component, and even a slight change in direction makes the vectors different.

2. Same Direction but Different Magnitudes: Similarly, if two vectors have the same direction but different magnitudes, they cannot be considered equal. The magnitude represents the length or size of the vector, and if it differs, the vectors are not equal.

3. Same Magnitude and Same Direction: When two vectors have both the same magnitude and the same direction, they are considered equal. This means that they represent the same physical quantity or displacement, even if they are located at different positions in space.

Therefore, the correct answer is option 'D': Two vectors are equal if and only if they have the same magnitude and direction. This condition ensures that all aspects of the vector, including both magnitude and direction, are identical.

D