1. Meaning of Like Vectors
Like vectors are vectors that point in the same direction. Their magnitudes may be equal or different, but their overall direction matches.
If two vectors have the same line of action and point the same way, they are considered like vectors.
1.1. Example of Like Vectors
Consider the vectors:
\[ \vec{u} = \langle 2, 4 \rangle, \quad \vec{v} = \langle 1, 2 \rangle \]
These two vectors point in the same direction since:
\[ \vec{u} = 2\vec{v} \]
They are multiples of each other, so they are like vectors.
2. Meaning of Unlike Vectors
Unlike vectors are vectors that point in different directions. Their magnitudes may be equal or different, but the direction does not match.
They do not lie on the same line, and they cannot be obtained by multiplying one by a positive scalar.
2.1. Example of Unlike Vectors
Consider the vectors:
\[ \vec{p} = \langle 2, 4 \rangle, \quad \vec{q} = \langle -1, 2 \rangle \]
These vectors point in different directions. One points toward the upper-right region, while the other points toward the upper-left region, so they are unlike.
3. Visual Interpretation
A simple way to identify like and unlike vectors is to look at the direction of their arrows. If the arrows point the same way, they are like. If not, they are unlike.
3.1. Key Idea
The magnitudes don't matter when deciding if vectors are like or unlike — only the direction matters.
4. More Examples
- \(\langle 3, 6 \rangle\) and \(\langle 0.5, 1 \rangle\) are like vectors.
- \(\langle 1, 0 \rangle\) and \(\langle 0, 1 \rangle\) are unlike vectors because they are perpendicular.
- \(\langle -2, -4 \rangle\) and \(\langle 1, 2 \rangle\) are like vectors because they point in exactly opposite directions but lie on the same line — however, they are anti-parallel, not unlike.
This makes it easier to distinguish between direction differences and opposite-but-collinear directions.