Two vectors are said to be equal if which of the following conditions is satisfied?

Introduction / Context:
In vector physics, quantities such as displacement, velocity and force are represented by vectors, which have both magnitude and direction. Defining when two vectors are equal is essential for adding vectors, comparing physical quantities and solving problems in mechanics and fields. This question checks whether you remember the precise condition for vector equality.

Given Data / Assumptions:

• We are working with geometric vectors in a plane or in space.
• Each vector has magnitude (length) and direction.
• The question asks for the condition under which two vectors are considered equal.
• Options distinguish between matching magnitudes, directions or both.

Concept / Approach:
By definition, two vectors are equal if they have the same magnitude and the same direction, regardless of their initial points or positions in space. Vectors can be moved parallel to themselves without changing their identity. If only magnitudes match but directions differ, they are not equal, even if one is opposite; they might be negatives of each other. If only directions match but lengths differ, they represent different physical quantities. Therefore, equality requires both equal length and equal direction.

Step-by-Step Solution:
Step 1: Recall that a vector is fully specified by magnitude and direction.Step 2: Understand that equality of vectors means complete coincidence of these two characteristics.Step 3: If only magnitudes are equal, vectors with different directions can represent quite different physical effects, so they are not equal.Step 4: If only directions are equal but magnitudes differ, the vectors are parallel but not equal.Step 5: If magnitudes are equal but directions are opposite, the vectors are negatives of each other, not equal.Step 6: Therefore, two vectors are equal only when both their magnitudes and directions are the same.

Verification / Alternative check:
Consider two displacement vectors representing walking 5 metres north from different starting points. Both have magnitude 5 metres and direction north, so they are equal as vectors even if they start at different locations. If one displacement were 5 metres north and the other 5 metres south, they would have equal magnitudes but opposite directions, representing different physical moves, so they would not be equal. This example illustrates that both magnitude and direction must match for vectors to be equal.

Why Other Options Are Wrong:
Option a, only magnitudes are same, ignores direction and would treat opposite vectors as equal, which is incorrect. Option b, only directions are same, would treat vectors of different lengths as equal, which is also wrong. Option d, magnitudes same but directions opposite, describes vectors that are equal in magnitude but negative of each other, not equal vectors.

Common Pitfalls:
Students sometimes confuse equal vectors with parallel vectors while doing vector diagrams. Parallel vectors have the same or opposite direction but can have different lengths. Equal vectors are a stricter concept requiring both same length and same direction. To avoid confusion, always check both properties when comparing vectors.

Final Answer:
Two vectors are said to be equal when both their magnitudes and directions are the same.