Collinearity of Two Vectors

Collinearity in the context of vectors refers to the condition where two vectors lie along the same line or parallel lines, regardless of their magnitudes or directions. When two vectors are collinear, one is a scalar multiple of the other. This concept is fundamental in vector algebra and has applications in various fields such as physics, engineering, and computer graphics.

Understanding Collinearity

To understand collinearity, let's consider two vectors $\vec{a}$ and $\vec{b}$. These vectors are said to be collinear if there exists a scalar $\lambda$ such that:

[ \vec{a} = \lambda \vec{b} ]

This equation implies that vector $\vec{a}$ can be obtained by stretching or compressing vector $\vec{b}$ by a factor of $\lambda$, and possibly reversing its direction if $\lambda$ is negative.

Conditions for Collinearity

For two vectors to be collinear, the following conditions must be met:

1. Proportional Components: The components of the vectors must be proportional to each other. If $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, then for collinearity, $a_1 = \lambda b_1$, $a_2 = \lambda b_2$, and $a_3 = \lambda b_3$.

2. Zero Cross Product: The cross product of two collinear vectors is the zero vector, $\vec{0}$. If $\vec{a} \times \vec{b} = \vec{0}$, then $\vec{a}$ and $\vec{b}$ are collinear.

3. Parallel or Anti-Parallel: Collinear vectors are either parallel or anti-parallel to each other. Parallel vectors have the same direction, while anti-parallel vectors have opposite directions.

Formulas Involving Collinearity

• Cross Product: $\vec{a} \times \vec{b} = \vec{0}$
• Scalar Multiplication: $\vec{a} = \lambda \vec{b}$

Examples

Let's consider a few examples to illustrate the concept of collinearity:

1. Example 1: Given $\vec{a} = (2, 4, 6)$ and $\vec{b} = (1, 2, 3)$, are these vectors collinear?

To check for collinearity, we can see if the components of $\vec{a}$ are proportional to the components of $\vec{b}$:

[ \frac{2}{1} = \frac{4}{2} = \frac{6}{3} = 2 ]

Since all ratios are equal, $\vec{a}$ is indeed collinear with $\vec{b}$, and $\vec{a} = 2\vec{b}$.

1. Example 2: Given $\vec{c} = (3, -6, 9)$ and $\vec{d} = (-1, 2, -3)$, are these vectors collinear?

Checking the ratios of the corresponding components:

[ \frac{3}{-1} = \frac{-6}{2} = \frac{9}{-3} = -3 ]

The ratios are equal, so $\vec{c}$ is collinear with $\vec{d}$, and $\vec{c} = -3\vec{d}$.

Table of Differences and Important Points

Property	Collinear Vectors	Non-Collinear Vectors
Direction	Same or opposite	Can be any direction
Cross Product	Zero vector	Non-zero vector
Scalar Multiplication	Exists a scalar $\lambda$ such that $\vec{a} = \lambda \vec{b}$	No such scalar exists
Proportional Components	Components are proportional	Components are not proportional
Geometric Representation	Lie on the same or parallel lines	Do not lie on the same line

Conclusion

Collinearity of vectors is a simple yet powerful concept in vector algebra. It helps in determining the relationship between vectors and is useful in solving geometric problems. Understanding the conditions for collinearity and being able to apply them to examples is essential for students and professionals working with vectors.