Collinear Vectors 

Collinear vectors are vectors that lie along the same straight line or parallel lines. They may point in the same or opposite directions but maintain alignment on a single axis. Collinearity implies that one vector is a scalar multiple of the other. These vectors are essential in geometry, physics, and vector algebra, especially for analyzing linear motion, force, and direction. Understanding collinear vectors helps in solving vector equations and verifying geometric relationships between points or lines.

1.0Collinear Vectors Definition

Collinear vectors are vectors that lie along the same straight line or on parallel lines. They may point in the same or opposite directions but still remain aligned along a single line.

In simple terms, collinear vectors either overlap or can be made to overlap by scaling. They have the same or exactly opposite directions.

2.0What is the Collinear Vector?

A collinear vector is one that is in the same direction or exact opposite direction of another vector. It means their directions are linearly dependent, and their cross product is zero (in 3D).

3.0Collinear Vectors Formula

To check whether two vectors $\vec{A}$ and $\vec{B}$ are collinear, use the collinear vectors formula:

If: $\vec{A}=kB$ for some scalar k

then $\vec{A}$ and $\vec{B}$ are collinear.

Alternatively, for 2D vectors $\vec{A}=(a_1,a_2)and\vec{B}=(b_1,b_2)$ :

Collinear if:  $\frac{a_1}{b_1}=\frac{a_2}{b_2}$

In 3D, check using cross product: $\vec{A}\times \vec{B}=\vec{0}$

4.0Collinear Vectors Conditions

Two vectors are collinear if:

1. They are scalar multiples of each other: $\vec{A}=k\vec{B}$
2. Their direction ratios are proportional
3. Their cross product is zero
4. They lie along the same or parallel line

5.0Collinear Vectors Properties

Here are important properties of collinear vectors:

• The angle between collinear vectors is either 0° (same direction) or 180° (opposite).
• The cross product of collinear vectors is always zero.
• The dot product is maximized or minimized depending on direction.
• If vectors are collinear, one can be written as a scalar multiple of the other.
• The area formed by two collinear vectors (as a parallelogram) is zero.

6.0Differentiate Between Collinear and Parallel Vectors

All collinear vectors are parallel, but not all parallel vectors are collinear in space.

7.0How Do You Know If Two Vectors Are Collinear?

Use any of these methods:

• Check if they are scalar multiples
• Compare the direction ratios
• Confirm the cross product is zero
• Use the determinant of their matrix in 2D to see if the area is zero

8.0When Are A and B Collinear Vectors?

Vectors $\vec{A}and\vec{B}$ are collinear if:

• $\vec{A}=k\vec{B}$ for some scalar k 
• The angle between them is 0° or 180°
• The cross product $\vec{A}\times \vec{B}=\vec{0}$

9.0Solved Examples on Collinear Vectors 

Example 1: Let $\vec{A}=(2,4)and\vec{B}=(1,2)$. Check whether they are collinear or not. 

Solution: 

Check:

$\frac{2}{1}=2,\frac{4}{2}=2\Rightarrow \text{same ratio}$

Hence, $\vec{A}and\vec{B}$ are collinear vectors.

Example 2: Let $\vec{A}=(3,6,9)and\vec{B}=(1,2,3)$. Check whether they are collinear or not. 

Solution: 

Check: $\vec{A}=3\vec{B}\Rightarrow Collinear$

Example 3: Are the vectors $\vec{A}=(4,6)and\vec{B}=(2,3)$ collinear?

Solution: 

Check the ratio of components: $\frac{4}{2}=2,\frac{6}{3}=2$

 Since both ratios are equal, $\vec{A}=2\vec{B}$⇒ Collinear.

Example 4: Check whether $\vec{A}=(3,6,9)and\vec{B}=(1,2,3)$ are collinear.

Solution:

Check if : $\vec{A}=k\vec{B}$

$\frac{3}{1}=3,\frac{6}{2}=3,\frac{9}{3}=3$

All components have the same ratio ⇒ $\vec{A}=3\vec{B}$⇒ Collinear

Example 5: Are vectors $\vec{A}=(1,2,3)and\vec{B}=(2,4,6)$ collinear?

Solution: 

Compute : $\vec{A}\times \vec{B}$

$\vec{A}\times \vec{B}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 3\\ 2& 4& 6\end{array}\right|=\vec{0}$

Cross product is zero ⇒ Collinear vectors

Example 6: Are $\vec{A}=(5,-10)and\vec{B}=(-2,4)$collinear?

Solution:

Check ratios: $\frac{5}{-2}=-2.5,\frac{-10}{4}=-2.5$

Both are equal ⇒ $\vec{A}=-2.5\vec{B}$⇒ Collinear

Example 7: If $\vec{A}=(k,6)and\vec{B}=(2,4)$ are collinear, find the value of k.

Solution:

Collinear condition: $\frac{k}{2}=\frac{6}{4}=\frac{3}{2}\Rightarrow k=2\cdot \frac{3}{2}=3$

 Answer: k = 3

10.0Practice Questions on Collinear Vectors

1. Are $\vec{A}=(2,-6)and\vec{B}=(-1,3)$ collinear?
2. Find if $\vec{A}=(4,8,12)and\vec{B}=(2,4,6)$ are collinear.
3. Prove that vectors $\vec{A}=(1,2,3)and\vec{B}=(2,4,6)$ are collinear.
4. Find the value of kk if (3, k) is collinear with (6, 8).
5. Determine whether $\vec{A}=(5,2)and\vec{B}=(15,6)$ are collinear.