Collinear Vectors

An Intuitive Understanding of Collinearity

The Necessary and Sufficient Condition for Collinearity

Methods for Testing Collinearity

1. The scalar-multiple method: check whether there is a real number $\lambda$ with $\vec{a} = \lambda\vec{b}$
2. The linear-combination method: check whether there exist $k, l$ not both zero with $k\vec{a} + l\vec{b} = \vec{0}$
3. The coordinate method: compute the ratio between the components of the vectors

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Worked Examples and Applications

A positive example: collinear vectors

Example 1: decide whether the vectors $\vec{a} = (3, 6)$ and $\vec{b} = (1, 2)$ are collinear.

Solution:

1. The scalar-multiple relation: $\vec{a} = 3\vec{b}$, so $\vec{a} = 3\vec{b}$
2. The linear combination: $1 \cdot \vec{a} + (-3) \cdot \vec{b} = 3(1,2) + (-3)(1,2) = (3,6) + (-3,-6) = (0,0)$
3. The ratio of the coordinates: the $x$-component ratio is $3\div1=3$ and the $y$-component ratio is $6\div2=3$; the ratios are equal

A negative example: non-collinear vectors

Example 2: decide whether the vectors $\vec{a} = (1, 2)$ and $\vec{b} = (2, 3)$ are collinear.

Solution:

1. The ratio of the coordinates: the $x$-component ratio is $1\div2=0.5$ and the $y$-component ratio is $2\div3\approx0.667$; the ratios differ
2. The linear combination: suppose $k(1,2) + l(2,3) = (0,0)$
   • the system: $k + 2l = 0$, $2k + 3l = 0$
   • solving gives $k = 0$, $l = 0$ (the only solution is the zero one)

The special case: the zero vector

Example 3: decide whether the zero vector $\vec{0}$ and the vector $\vec{a} = (2, 5)$ are collinear.

Solution:

• The linear combination: $1 \cdot \vec{0} + 0 \cdot \vec{a} = (0,0)$
• The coefficients $(1, 0)$ are not both zero

Application Scenarios

Applications in spatial geometry

1. Equations of lines in space: a line through two points in space can be described by using collinear vectors to represent the position vector of any point on the line
2. Testing linear combinations: collinearity of vectors is the foundation for studying linear dependence
3. Geometric transformations: collinear vectors play an important role in transformations such as translation and scaling

A reminder of common misconceptions

Practice

Exercise 1

Decide whether the vectors $\vec{a} = (4, -2)$ and $\vec{b} = (-2, 1)$ are collinear.

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Summary

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