1. Zero Vector
A zero vector is a vector whose magnitude is zero. It has no specific direction because all its components are zero.
In coordinate form, it is written as:
\[ \vec{0} = \langle 0, 0 \rangle \quad \,\text{or}\,\quad \langle 0, 0, 0 \rangle \]
The zero vector does not point anywhere; it simply represents ‘no movement’ or ‘no change.’
1.1. Example
If a point does not move from its position, the displacement is the zero vector \(\vec{0}\).
2. Unit Vector
A unit vector is a vector of magnitude 1. It is used to represent direction without scaling.
To form a unit vector \(\hat{v}\) from any vector \(\vec{v}\), divide it by its magnitude:
\[ \hat{v} = \dfrac{\vec{v}}{|\vec{v}|} \]
2.1. Example
For \( \vec{v} = \langle 3, 4 \rangle \), the unit vector is:
\[ \hat{v} = \left\langle \dfrac{3}{5}, \dfrac{4}{5} \right\rangle \]
3. Like and Unlike Vectors
Like vectors are vectors that have the same direction. Their magnitudes may differ.
Unlike vectors have different directions.
3.1. Example
The vectors \(\langle 2, 4 \rangle\) and \(\langle 1, 2 \rangle\) are like vectors because they point in the same direction.
But \(\langle 2, 4 \rangle\) and \(\langle -1, 2 \rangle\) are unlike because their directions differ.
4. Collinear and Non-Collinear Vectors
Collinear vectors lie along the same line or parallel lines.
Non-collinear vectors do not lie along the same line and cannot be placed parallel to one another.
4.1. Example
Vectors \(\langle 3, 6 \rangle\) and \(\langle -1, -2 \rangle\) are collinear because one is a multiple of the other.
The vectors \(\langle 1, 0 \rangle\) and \(\langle 0, 1 \rangle\) are non-collinear because they are perpendicular.
5. Parallel and Anti-Parallel Vectors
Parallel vectors have the same or exactly opposite direction but are not necessarily equal in magnitude.
Anti-parallel vectors are parallel but point in opposite directions.
5.1. Example
\(\langle 4, 2 \rangle\) and \(\langle 2, 1 \rangle\) are parallel.
\(\langle 4, 2 \rangle\) and \(\langle -2, -1 \rangle\) are anti-parallel.
6. Co-Initial and Co-Terminal Vectors
Co-initial vectors start from the same point.
Co-terminal vectors end at the same point.
Their magnitudes and directions may differ; this classification is based purely on reference points.
6.1. Example
If vectors \(\vec{u}\) and \(\vec{v}\) start at the same point A, they are co-initial.
If two vectors both end at point B, they are co-terminal.
7. Examples of Different Types of Vectors
Some quick examples that show different vector types:
- \(\langle 0, 0, 0 \rangle\) — zero vector
- \(\langle \tfrac{1}{\sqrt{3}}, \tfrac{1}{\sqrt{3}}, \tfrac{1}{\sqrt{3}} \rangle\) — unit vector
- \(\langle 5, 10 \rangle\) and \(\langle 1, 2 \rangle\) — like vectors
- \(\langle 4, 0 \rangle\) and \(\langle 0, 3 \rangle\) — non-collinear
- \(\langle 3, -6 \rangle\) and \(\langle -1, 2 \rangle\) — anti-parallel