Table of Content

• What are Vectors?
• What are Collinear Vectors?
• Conditions for collinearity of vectors
• Collinear Vs Parallel Vectors
• Solved Examples
• Practice Problems
• FAQs: Collinear Vectors

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,

• Types of Vectors
• Scalars and Vectors
• Dot and Cross Products on Vectors
• Vector Operations

Solved Examples on Collinear Vectors

Question 1: Determine if [Tex]\overrightarrow{a} = \{1,5\}, \overrightarrow{b} = \{3,15\}[/Tex] are collinear to each other?

Solution:

Given [Tex]\overrightarrow{a} = \{1,5\}, \overrightarrow{b} = \{3,15\}[/Tex]

As [Tex] 3\{1,5\} = \{3,15\}[/Tex],

[Tex]\overrightarrow{a} = 3\overrightarrow{b}[/Tex]

Thus a and b are collinear to each other.

Question 2: Are [Tex]\overrightarrow{a} = \{1,2\}, \overrightarrow{b} = \{3,6\}, \overrightarrow{c} = \{4,5\}[/Tex] collinear or not?

Solution:

Given [Tex]\overrightarrow{a} = \{1,2\}, \overrightarrow{b} = \{3,6\}, \overrightarrow{c} = \{4,5\}[/Tex]

As [Tex]\frac{1}{3} = \frac{2}{6}[/Tex], [Tex]\overrightarrow{a}[/Tex] is collinear to [Tex]\overrightarrow{b}[/Tex].

Now [Tex]\frac{3}{4} = \frac{6}{5}[/Tex], [Tex]\overrightarrow{b}[/Tex] is not collinear to [Tex]\overrightarrow{c}[/Tex] and in turn [Tex]\overrightarrow{c}[/Tex] is not collinear to [Tex]\overrightarrow{a}[/Tex].

Question 3: Check using cross product if [Tex]\overrightarrow{p} = 2\hat{i}+3\hat{j}+4\hat{k} [/Tex]and [Tex]\overrightarrow{q} = 8\hat{i}+3\hat{j}+1\hat{k} [/Tex]are collinear?

Solution:

Given [Tex]\overrightarrow{p} = 2\hat{i}+3\hat{j}+4\hat{k} [/Tex]and [Tex]\overrightarrow{q} = 8\hat{i}+3\hat{j}+1\hat{k} [/Tex]

[Tex]\overrightarrow{p} \times \overrightarrow{q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ 2 & 3 & 4\\ 8 & 3 & 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[/Tex]

Thus, given vectors are not collinear.

Question 4: Are [Tex]\overrightarrow{p} = 4\hat{i}+1\hat{j}+0\hat{k} [/Tex]and [Tex]\overrightarrow{t} = 1\hat{i}+8\hat{j}+9\hat{k}[/Tex] collinear or not?

Solution:

Given, [Tex]\overrightarrow{p} = 4\hat{i}+1\hat{j}+0\hat{k} [/Tex]and [Tex]\overrightarrow{t} = 1\hat{i}+8\hat{j}+9\hat{k}[/Tex]

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 [Tex]\overrightarrow{p} = 7\hat{i}+4\hat{j}+8\hat{k}[/Tex] and [Tex]\overrightarrow{q} = 3\hat{i}+9\hat{j}+3\hat{k}[/Tex] are collinear?

Solution:

Given [Tex]\overrightarrow{p} = 7\hat{i}+4\hat{j}+8\hat{k}[/Tex] and [Tex]\overrightarrow{q} = 3\hat{i}+9\hat{j}+3\hat{k} [/Tex]

[Tex]\overrightarrow{p} \times \overrightarrow{q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}\\ 7 & 4 & 8\\ 3 & 9 & 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[/Tex]

Thus, given vectors are not collinear.

Practice Problems on Collinear Vectors

Problem 1: Determine if [Tex]\overrightarrow{a} = \{1,8\}, \overrightarrow{b} = \{2,9\}[/Tex] are collinear to each other?

Problem 2: Are [Tex]\overrightarrow{a} = \{8,6\}, \overrightarrow{b} = \{9,4\}, \overrightarrow{c} = \{4,5\}[/Tex] collinear or not?

Problem 3: Check using cross product if [Tex]\overrightarrow{p} =1\hat{i}+5\hat{j}+8\hat{k}[/Tex] and [Tex]\overrightarrow{q} = 3\hat{i}+8\hat{j}+3\hat{k}[/Tex] are collinear?

Problem 4: Check using cross product if [Tex]\overrightarrow{p} = 2\hat{i}+1\hat{j}+5\hat{k}[/Tex] and [Tex]\overrightarrow{q} = 9\hat{i}+0\hat{j}+1\hat{k}[/Tex] are collinear?

Problem 5: Are [Tex]\overrightarrow{p} = 12\hat{i}+8\hat{j}+5\hat{k}[/Tex] and [Tex]\overrightarrow{t} = 19\hat{i}+6\hat{j}+4\hat{k}[/Tex] collinear or not?

FAQs: Collinear Vectors

What is Collinearity?

Collinearity is a geometric property where points, vectors, or other geometric entities lie on the same straight line. For vectors, they are collinear if one vector can be expressed as a scalar multiple of another, meaning they have the same or opposite direction and differ only in magnitude.

How to know if vectors are collinear?

To determine if vectors are collinear, check if one vector is a scalar multiple of the other, or use the cross product method—the vectors are collinear if their cross product is zero.

What is the formula for collinear three vectors?

To check if three vectors are collinear, you can use the condition derived from the cross product. Specifically, for vectors ????a, ????b, and ????c, you can check collinearity by confirming that the scalar triple product is zero i.e., a⋅(b×c) = 0.

What is the area of triangle formed by 3 collinear vectors?

We know that area formed by any 3 collinear lines is zero. As vectors are also lines with directions, area formed by any 3 collinear vectors is also zero.

Are collinear vectors always equal in magnitude?

It is not necessary for collinear vectors to be equal in magnitude as two vectors are said to be collinear if one vector is a scalar multiple of the other vector.