1. Meaning of Co-Initial Vectors
Co-initial vectors are vectors that start from the same initial point. Their magnitudes and directions may be different, but all of them begin at one common point.
The shared starting point is what makes them co-initial, not their direction or length.
1.1. Example of Co-Initial Vectors
Imagine two vectors drawn from the same point A:
\[ \vec{u} = \overrightarrow{AB}, \quad \vec{v} = \overrightarrow{AC} \]
Both vectors start at A, so they are co-initial. They may point in different directions, but their origin is the same.
2. Meaning of Co-Terminal Vectors
Co-terminal vectors are vectors that end at the same final point. Their starting points can be different, but their ending point is common.
Even if they approach the ending point from different directions, they are still co-terminal because their terminal point is the same.
2.1. Example of Co-Terminal Vectors
Consider vectors ending at point B:
\[ \vec{p} = \overrightarrow{AB}, \quad \vec{q} = \overrightarrow{CB} \]
Here, both vectors end at B, so they are co-terminal.
3. Difference Between Co-Initial and Co-Terminal Vectors
The idea is simple: co-initial vectors share the same starting point, while co-terminal vectors share the same ending point.
Their magnitudes and directions do not need to match in either case.
3.1. Quick Comparison
- Co-initial: same origin, different or same directions.
- Co-terminal: same ending point, may start anywhere.
4. Geometric Interpretation
On a diagram, co-initial vectors look like multiple arrows starting from the same point and spreading out in different directions. Co-terminal vectors look like arrows coming in toward the same point from different places.
4.1. Key Idea
Co-initial → same start.
Co-terminal → same end.
Direction and magnitude do not affect this classification.
5. More Examples
- \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\): co-initial (same O).
- \(\overrightarrow{AP}\) and \(\overrightarrow{BP}\): co-terminal (same P).
- \(\langle 3, 1 \rangle\) from different starting points can still be co-terminal if their endpoints coincide.