1. Meaning of Parallel Vectors
Parallel vectors are vectors that have the same or exactly opposite direction but lie along parallel lines. They share the same line of action or can be placed on lines that never intersect.
A key idea: if one vector is a scalar multiple of another, the two vectors are parallel.
For vectors \(\vec{u}\) and \(\vec{v}\), they are parallel if:
\[ \vec{u} = k\vec{v} \]
for some real number \(k\). When \(k > 0\), they point the same way.
1.1. Example of Parallel Vectors
Consider:
\[ \vec{u} = \langle 4, 2 \rangle, \quad \vec{v} = \langle 2, 1 \rangle \]
Since:
\[ \vec{u} = 2\vec{v} \]
both vectors point in the same direction and are parallel.
2. Meaning of Anti-Parallel Vectors
Anti-parallel vectors are parallel vectors that point in exactly opposite directions.
This happens when one vector is a negative scalar multiple of the other.
So, \(\vec{u}\) and \(\vec{v}\) are anti-parallel if:
\[ \vec{u} = k\vec{v} \quad \text{with} \; k < 0 \]
They lie on the same line or on parallel lines, but their arrows face opposite ways.
2.1. Example of Anti-Parallel Vectors
Take the vectors:
\[ \vec{p} = \langle 4, 2 \rangle, \quad \vec{q} = \langle -2, -1 \rangle \]
Here:
\[ \vec{p} = -2\vec{q} \]
They point in opposite directions, so they are anti-parallel.
3. Geometric Interpretation
Parallel vectors lie along the same direction or the same straight path when placed tail-to-tail. Anti-parallel vectors lie along the same line but face opposite directions.
Even if their magnitudes are very different, only direction matters here.
3.1. Key Idea
Parallel → same direction.
Anti-parallel → opposite direction.
Non-parallel → neither of the above.
4. How to Check If Two Vectors Are Parallel
To test if two vectors are parallel, compare their component ratios.
For \(\vec{a} = \langle a_1, a_2 \rangle\) and \(\vec{b} = \langle b_1, b_2 \rangle\), they are parallel if:
\[ \dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} \]
In 3D, check:
\[ \dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} = \dfrac{a_3}{b_3} \]
4.1. Example Check
For vectors \(\langle 3, 9, -6 \rangle\) and \(\langle 1, 3, -2 \rangle\):
Ratios:
\[ 3:1 = 9:3 = -6:(-2) = 3 \]
Since all ratios match, the vectors are parallel.
5. More Examples
- \(\langle 5, 10 \rangle\) and \(\langle 1, 2 \rangle\): parallel.
- \(\langle -5, -10 \rangle\) and \(\langle 1, 2 \rangle\): anti-parallel.
- \(\langle 2, 0 \rangle\) and \(\langle 0, 2 \rangle\): non-parallel (perpendicular).