Collinearity of Three Vectors

Collinearity in the context of vectors refers to the condition where three or more vectors lie on the same line or, in other words, are parallel to the same line. When dealing with vectors in a two-dimensional or three-dimensional space, understanding collinearity is crucial as it has implications in various applications such as physics, engineering, and computer graphics.

Understanding Vectors

Before diving into collinearity, let's briefly review what vectors are. A vector is a mathematical entity that has both magnitude (length) and direction. Vectors are often represented graphically by arrows, where the length of the arrow indicates the magnitude and the direction in which the arrow points indicates the direction of the vector.

Conditions for Collinearity

For three vectors to be collinear, they must satisfy certain conditions. These conditions can be expressed in terms of vector arithmetic or geometric interpretation.

Vector Arithmetic Approach

Three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are collinear if there exist scalars $k$ and $m$ such that:

[ \vec{c} = k\vec{a} + m\vec{b} ]

where $k$ and $m$ are not both zero, and $\vec{a}$ and $\vec{b}$ are not parallel (not collinear).

Geometric Interpretation

Geometrically, three vectors are collinear if they lie along the same line or if one of the vectors can be expressed as a linear combination of the other two.

Determining Collinearity

To determine if three vectors are collinear, we can use the following methods:

Method 1: Cross Product

For vectors in three-dimensional space, the cross product can be used to determine collinearity. If the cross product of any two of the three vectors is the zero vector, then the vectors are collinear.

[ \vec{a} \times \vec{b} = \vec{0} ]

Method 2: Ratio of Components

If the corresponding components of the vectors are proportional, then the vectors are collinear. For vectors $\vec{a} = (a_1, a_2, a_3)$, $\vec{b} = (b_1, b_2, b_3)$, and $\vec{c} = (c_1, c_2, c_3)$, the condition for collinearity is:

[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} ]

Method 3: Linear Dependence

Vectors are collinear if they are linearly dependent. This means that one of the vectors can be written as a linear combination of the others.

Table of Differences and Important Points

Property	Description
Definition	Collinearity refers to vectors lying on the same line.
Conditions	Vectors are collinear if they are proportional or one is a linear combination of the others.
Determination	Use cross product, ratio of components, or linear dependence to determine collinearity.
Cross Product	Zero cross product indicates collinearity in 3D space.
Ratio of Components	Proportional components indicate collinearity.
Linear Dependence	If vectors are linearly dependent, they are collinear.

Examples

Example 1: Using Cross Product

Let's consider three vectors in 3D space:

[ \vec{a} = (1, 2, 3), \quad \vec{b} = (2, 4, 6), \quad \vec{c} = (3, 6, 9) ]

To check for collinearity, we can compute the cross product of $\vec{a}$ and $\vec{b}$:

[ \vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 2 & 3 \ 2 & 4 & 6 \ \end{vmatrix} = (12 - 12)\mathbf{i} - (6 - 6)\mathbf{j} + (4 - 4)\mathbf{k} = \vec{0} ]

Since the cross product is the zero vector, $\vec{a}$ and $\vec{b}$ are collinear. Since $\vec{c}$ is also a scalar multiple of $\vec{a}$ and $\vec{b}$, all three vectors are collinear.

Example 2: Using Ratio of Components

Consider the vectors:

[ \vec{a} = (1, 3, 4), \quad \vec{b} = (2, 6, 8), \quad \vec{c} = (3, 9, 12) ]

The ratios of the corresponding components are:

[ \frac{1}{2} = \frac{3}{6} = \frac{4}{8} \quad \text{and} \quad \frac{1}{3} = \frac{3}{9} = \frac{4}{12} ]

Since the ratios are equal, the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are collinear.

In conclusion, understanding the concept of collinearity of vectors is essential for various fields that utilize vector analysis. By using the methods outlined above, one can determine whether three vectors are collinear, which is a fundamental aspect of vector geometry.