Vectors are also called Euclidean vectors or Spatial vectors, and they have many applications in mathematics, physics, engineering, and various other fields. There are different types of vectors, including zero vectors (which have 0 magnitude and no direction), unit vectors (which have a magnitude of 1), position vectors, co-initial vectors, like and unlike vectors, co-planar vectors, collinear vectors, equal vectors, displacement vectors, and negative vectors.
In this article, we will discuss collinear vectors and the criteria according to which two vectors are said to be collinear in detail.
What are Vectors?
A vector is a mathematical entity that has both magnitude (amount of movement) and direction. It is used to represent physical quantities like distance, velocity, acceleration, force, and more. Vectors are geometric entities that can be represented by a line with an arrow pointing towards its direction, and its length represents the magnitude of the vector.
There are many types of vectors based on various different properties, such as:
- Unit Vector
- Orthogonal Vector
- Parallel Vector
- Anti-parallel Vector
- Zero Vectors
- Negative of a Vector
- Equal Vectors
- Collinear Vectors
- Coplanar Vectors
- Position Vectors
- Displacement Vectors
- Localized Vectors
- Non-localized Vectors
- Co-initial Vectors
- Like and Unlike Vectors
In this article, we will be discussing the concept of "Collinear Vectors" in detail.
What are Collinear Vectors?
Collinear vectors are vectors that lie along the same line or are parallel to the same line, regardless of their magnitude or direction. This implies that one vector can be expressed as a scalar multiple of the other.
In simpler terms, if you have two vectors, a and b, they are collinear if there exists a scalar k such that a = kb.
The concept of collinearity is important in various mathematical and physical contexts. For instance, in geometry, it helps determine if points are aligned in a straight line. In physics, collinear vectors helps us analyze forces and motion, as it helps simplify problems where multiple forces are acting along the same line.
Visual Representation of Collinear Vectors
Visual representation of collinear vectors can be quite straightforward and intuitive. Imagine plotting vectors on a graph. If vectors are collinear, they would appear as arrows that lie either on the same straight line or parallel to same straight line, whether they point in the same direction or in opposite directions.
Conditions for Collinearity of Vectors
We have learnt that two or more vectors that point in same or opposite directions and parallel to each other are called collinear vectors.
In mathematics, we have certain conditions that must be satisfied by two or more vectors to be considered collinear. Consider two vectors \overrightarrow{A} and \overrightarrow{B} .The conditions for collinearity of vectors are as follows:
Condition 1: If \overrightarrow{A} = n \overrightarrow{B}, where n is any scalar then vectors A and B are said to be collinear.
Condition 2: If the ratio of the corresponding coordinates of two vectors are equal, then they are said to be collinear. This condition is not applicable if any one of the coordinates of any vector is zero. Consider \overrightarrow{A} = a\hat{i}+b\hat{j}+c\hat{k} and \overrightarrow{B} = p\hat{i}+q\hat{j}+r\hat{k} , then they are said to be collinear if:
\bold{\frac{a}{p}=\frac{b}{q}=\frac{c}{r}} OR \bold{\frac{p}{a}=\frac{q}{b}=\frac{r}{c}}
Condition 3: Two vectors are said to be collinear if their cross product is zero i.e. \overrightarrow{A} \times \overrightarrow{B} = 0.
Read More about Collinear Points.
Collinear Vs Parallel Vectors
Parallel vectors are specific case of collinear vectors. Some other differences between collinear vectors and parallel vectors are listed in the following table:
| Feature | Collinear Vectors | Parallel Vectors |
|---|
| Definition | Vectors that lie along the same line | Vectors that have the same or opposite direction |
|---|
| Direction | May have the same or opposite direction | Always have the same or exactly opposite direction |
|---|
| Mathematical Relation | u =kv for some scalar k. | u = kv for some scalar k, k > 0 for same direction, k < 0 for opposite |
|---|
| Example | u = (1,2), and v = (2,4) | u =(3, 3), v =(−6,−6) (opposite direction) |
|---|
| Geometric Interpretation | Vectors that can be scaled to overlap when plotted from a common point | Vectors that are either exactly aligned or directly opposite when plotted from a common point |
|---|
Read More,
Solved Examples on Collinear Vectors
Question 1: Determine if \overrightarrow{a} = \{1,5\}, \overrightarrow{b} = \{3,15\} are collinear to each other?
Solution:
Given \overrightarrow{a} = \{1,5\}, \overrightarrow{b} = \{3,15\}
As 3\{1,5\} = \{3,15\},
\overrightarrow{a} = 3\overrightarrow{b}
Thus a and b are collinear to each other.
Question 2: Are \overrightarrow{a} = \{1,2\}, \overrightarrow{b} = \{3,6\}, \overrightarrow{c} = \{4,5\} collinear or not?
Solution:
Given \overrightarrow{a} = \{1,2\}, \overrightarrow{b} = \{3,6\}, \overrightarrow{c} = \{4,5\}
As \frac{1}{3} = \frac{2}{6}, \overrightarrow{a} is collinear to \overrightarrow{b}.
Now \frac{3}{4} = \frac{6}{5}, \overrightarrow{b} is not collinear to \overrightarrow{c} and in turn \overrightarrow{c} is not collinear to \overrightarrow{a}.
Question 3: Check using cross product if \overrightarrow{p} = 2\hat{i}+3\hat{j}+4\hat{k} and \overrightarrow{q} = 8\hat{i}+3\hat{j}+1\hat{k} are collinear?
Solution:
Given \overrightarrow{p} = 2\hat{i}+3\hat{j}+4\hat{k} and \overrightarrow{q} = 8\hat{i}+3\hat{j}+1\hat{k}
\overrightarrow{p} \times \overrightarrow{q} = \begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k}\\ 2 &amp; 3 &amp; 4\\ 8 &amp; 3 &amp; 1 \end{vmatrix}\\ = \hat{i}(3-12)-\hat{j}(2-32)+\hat{k}(6-24)\\ = -9\hat{i} +30\hat{j}-18\hat{k} \ne \overrightarrow0
Thus, given vectors are not collinear.
Question 4: Are \overrightarrow{p} = 4\hat{i}+1\hat{j}+0\hat{k} and \overrightarrow{t} = 1\hat{i}+8\hat{j}+9\hat{k} collinear or not?
Solution:
Given, \overrightarrow{p} = 4\hat{i}+1\hat{j}+0\hat{k} and \overrightarrow{t} = 1\hat{i}+8\hat{j}+9\hat{k}
As one of the vectors contain a zero coordinate, the two vectors are not collinear to each other.
Question 5: Check using cross product if \overrightarrow{p} = 7\hat{i}+4\hat{j}+8\hat{k} and \overrightarrow{q} = 3\hat{i}+9\hat{j}+3\hat{k} are collinear?
Solution:
Given \overrightarrow{p} = 7\hat{i}+4\hat{j}+8\hat{k} and \overrightarrow{q} = 3\hat{i}+9\hat{j}+3\hat{k}
\overrightarrow{p} \times \overrightarrow{q} = \begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k}\\ 7 &amp; 4 &amp; 8\\ 3 &amp; 9 &amp; 3 \end{vmatrix}\\ = \hat{i}(12-72)-\hat{j}(21-24)+\hat{k}(63-12)\\ = -60\hat{i} +3\hat{j}+51\hat{k} \ne\overrightarrow0
Thus, given vectors are not collinear.
Practice Problems on Collinear Vectors
Problem 1: Determine if \overrightarrow{a} = \{1,8\}, \overrightarrow{b} = \{2,9\} are collinear to each other?
Problem 2: Are \overrightarrow{a} = \{8,6\}, \overrightarrow{b} = \{9,4\}, \overrightarrow{c} = \{4,5\} collinear or not?
Problem 3: Check using cross product if \overrightarrow{p} =1\hat{i}+5\hat{j}+8\hat{k} and \overrightarrow{q} = 3\hat{i}+8\hat{j}+3\hat{k} are collinear?
Problem 4: Check using cross product if \overrightarrow{p} = 2\hat{i}+1\hat{j}+5\hat{k} and \overrightarrow{q} = 9\hat{i}+0\hat{j}+1\hat{k} are collinear?
Problem 5: Are \overrightarrow{p} = 12\hat{i}+8\hat{j}+5\hat{k} and \overrightarrow{t} = 19\hat{i}+6\hat{j}+4\hat{k} collinear or not?
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