1. Meaning of Collinear Vectors
Collinear vectors are vectors that lie along the same straight line or along parallel lines. They may point in the same direction or in opposite directions, but they must share the same line of action.
Another way to identify collinearity is: if one vector is a scalar multiple of another, then the two vectors are collinear.
For vectors \(\vec{u}\) and \(\vec{v}\), they are collinear if:
\[ \vec{u} = k \vec{v} \]
for some real number \(k\).
1.1. Example of Collinear Vectors
Take the vectors:
\[ \vec{u} = \langle 3, 6 \rangle, \quad \vec{v} = \langle 1, 2 \rangle \]
Since:
\[ \vec{u} = 3\vec{v} \]
they lie on the same line and are collinear.
If one of them were negative, like \(\langle -1, -2 \rangle\), they would still be collinear but in opposite directions.
2. Meaning of Non-Collinear Vectors
Non-collinear vectors do not lie on the same line and cannot be made parallel by scaling. Their directions differ in such a way that no scalar multiple links them.
These vectors can span a plane, form angles with each other, and generally point in independent directions.
2.1. Example of Non-Collinear Vectors
Consider the vectors:
\[ \vec{p} = \langle 1, 0 \rangle, \quad \vec{q} = \langle 0, 1 \rangle \]
These vectors are perpendicular. Since neither is a multiple of the other, they are non-collinear.
3. How to Check Collinearity
To test if two vectors are collinear, compare their components. If the ratios of their corresponding components are equal, the vectors are collinear.
Given \(\vec{a} = \langle a_1, a_2 \rangle\) and \(\vec{b} = \langle b_1, b_2 \rangle\), the condition is:
\[ \dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} \]
For 3D vectors, the idea is similar.
3.1. Example Check
For vectors \(\langle 4, 8, 12 \rangle\) and \(\langle 1, 2, 3 \rangle\):
All ratios are:
\[ 4:1 = 8:2 = 12:3 = 4 \]
So the vectors are collinear.
4. Visual Understanding
One way to picture collinearity is to imagine arrows drawn on a sheet of paper. If they lie along the same straight path or parallel tracks, they are collinear. If they form any angle between them (other than zero or 180°), they are non-collinear.
4.1. Key Idea
Collinear → same line or parallel lines.
Non-collinear → different lines with an angle between them.
5. More Examples
- \(\langle 2, 4 \rangle\) and \(\langle -1, -2 \rangle\) — collinear (opposite directions).
- \(\langle 5, 0 \rangle\) and \(\langle 0, 7 \rangle\) — non-collinear (perpendicular).
- \(\langle 3, 9, 12 \rangle\) and \(\langle 1, 3, 4 \rangle\) — non-collinear (ratios don’t match).