Vectors are quantities that have both magnitude (size) and direction. They show how and where an object moves from one place to another. For example, if something moves 5 meters to the east, that movement is a vector. Vectors help us understand motion and force in physics and math. There are different types of vectors, such as:

• Collinear vectors (point in the same or opposite direction),
• Zero vector (no length),
• Unit vector (length of 1),
• Position vector,
• Equal vectors,
• Like and unlike vectors, and
• Negative vectors (opposite direction).

What are Collinear Vectors?

Collinear vectors can be defined as when two or more than two vectors are parallel to one another irrespective of the magnitude or the direction. If we consider vectors as a straight line, then the extent of the line denotes the magnitude, and the pointer on this line is the direction in which the vector is traveling. Therefore, we can estimate any two given vectors in vector algebra as collinear if and only if these two vectors are either along the identical line or the vectors are parallel to one another in the same/opposite direction.

The parallel nature of vectors indicates that they never meet or intersect with each other. Therefore collinear vectors are also known as parallel vectors.

Collinear Vectors Formula

Two parallel vectors are collinear due to the reason that these two vectors are indicating in an identical direction or opposing direction. If two vectors are represented as \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\text{ and }\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\)

Two vectors are collinear if and only if \(\frac{b_1}{a_1}=\frac{b_2}{a_2}=\frac{b_3}{a_3}\)

Condition for Collinear Vectors

For any 2 vectors to be collinear vectors, they have to fulfill the given conditions.

Condition 1: Two vectors \(\vec{a}\text{ and }\vec{b}\) are said to be collinear if there exists a nonzero scalar ‘n’ such that:

\(\vec{b}=n\vec{a}\)

Condition 2: Two vectors \(\vec{a}\text{ and }\vec{b}\) are supposed to be collinear if and only if the proportion of their related coordinates is identical. That is if:

\(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\text{ and }\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) are two vectors:

Then

\(b_1\hat{i}+b_2\hat{j}+b_3\hat{k}=n\left(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\right)\)

\(b_1=na_1\text{,}\ b_2=na_2\text{ and }\ b_3=na_3\)

Condition 2 is not correct if any one of the elements of the provided vector is analogous to zero.

Condition 3: Two vectors \(\vec{a}\text{ and }\vec{b}\) are also said to be collinear if their cross product is equivalent to a zero vector. That is if \(\vec{a}\text{ and }\vec{b}\) are two nonzero vectors. Then \(\vec{a}\times\vec{b}=\vec{0}\) if and only if \(\vec{a}\text{ and }\vec{b}\) are parallel or collinear to each other, i.e.,\(\vec{a}\times\vec{b}=\vec{0}\Leftrightarrow\vec{a}\text{ || }\vec{b}\).

Condition 3 can be used only in 3D or spatial situations.

How to Prove Collinearity of Vectors

Now let us learn how to prove that two vectors are collinear vectors or how to prove the collinearity of vectors. For the two given vectors to be collinear we can check for the three important conditions as discussed above. Let us take an example to understand the same.

To check for the collinearity we can obtain the cross-product between them as shown:

If \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\text{ and }\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) are the two vectors. Then:

\(\vec{a}\times\vec{b}=\begin{vmatrix}i&j&k

a_1&a_2&a_3

b_1&b_2&b_3\end{vmatrix}\)

\(=i\left(a_2b_3-a_3b_2\right)-j\left(a_1b_3-a_3b_1\right)+k\left(a_1b_2-a_2b_1\right)\)

\(=i\left(a_2na_3-a_3na_2\right)-j\left(a_1na_3-a_3na_1\right)+k\left(a_1na_2-a_2na_1\right)\)

\(\vec{a}\times\vec{b}=i\left(0\right)-j\left(0\right)+k\left(0\right)=\vec{0}\)

The resultant is zero as different components of vector of the identical vector are perpendicular to one another and therefore their product is zero.

Properties of Collinear Vectors

Same Line or Parallel Lines
Collinear vectors lie along the same line or in parallel directions. They may point in the same or opposite direction.

Proportional Components
If two vectors are collinear, then the ratio of their corresponding components (x, y, and z) is the same.
Example: If A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), then
a₁/b₁ = a₂/b₂ = a₃/b₃

Scalar Multiple
One vector is a scalar multiple of the other. That means:
B = k × A, where k is a real number.

Zero Cross Product
The cross product of two collinear vectors is zero.
That is: A × B = 0

Same or Opposite Direction
If the scalar k is positive, vectors point in the same direction.
If k is negative, they point in opposite directions.

Coplanar by Default
All collinear vectors are also coplanar (they lie on the same plane).

Here are some simple points to remember about collinear vectors:

• Two vectors are called collinear if they lie along the same line or are parallel to the same line.
• This means that one vector can be written as a multiple of the other.
• So, if two vectors point in the same direction (or exactly opposite), they are collinear—even if they have different lengths.
• In short, two vectors are collinear if they follow the same path or direction in space.

Solved Examples of Collinear Vectors

Let us practice some solved examples of collinear vectors for more clarity.

Example 1: Which of the given vectors \(\vec{a}\)= {2, 3}, \(\vec{b}\)= {4, 6}, \(\vec{c}\)= {6, 12} are collinear to one another?

Solution: To check for collinearity we will use the below condition:

If \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\text{ and }\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) then:

\(\frac{b_1}{a_1}=\frac{b_2}{a_2}=\frac{b_3}{a_3}\) for the collinearity of vectors.

Consider: \(\vec{a}\text{ and }\vec{b}\)

Given: \(\vec{a}\)= {2, 3} and \(\vec{b}\)= {4, 6}

\(\frac{2}{4}=\frac{3}{6}=\frac{1}{2}\)

Hence vectors a and b are collinear.

Now, consider

\(\vec{a}\text{ and }\vec{c}\)

\(\vec{a}\)= {2, 3} and \(\vec{c}\)= {6, 12}

\(\frac{2}{6}=\frac{3}{12}\Rightarrow\frac{1}{3}\ne\frac{1}{4}\)

Hence vectors a and c are non-collinear.

Now, consider

\(\vec{b}\text{ and }\vec{c}\)

\(\vec{b}\)= {4, 6} and \(\vec{c}\)= {6, 12}

\(\frac{4}{6}=\frac{6}{12}\Rightarrow\frac{2}{3}\ne\frac{1}{2}\)

Hence vectors b and c are non-collinear.

Example 2: Show that the vectors \(\vec{a}=\left(3,5,7\right)\text{,}\vec{b}=\left(6,10,14\right)\) are collinear vectors.

Solution: To check if vectors are collinear we will apply conditions for collinear vectors.

For \(\vec{a}=\left(3,5,7\right)\text{,}\vec{b}=\left(6,10,14\right)\)

\(\frac{3}{6}=\frac{5}{10}=\frac{7}{14}\Rightarrow\frac{1}{2}=\frac{1}{2}=\frac{1}{2}\)

Hence vectors are collinear.

Example 3: Find the value of n that makes the vectors a = (2, 5) and b = (4, n) collinear.

Solution
Two vectors are collinear when one is a scalar multiple of the other.
Let b = k · a.

1. Compare the first components:
   4 = k · 2  ⇒ k = 2
2. Use the same k for the second components:
   n = k · 5 = 2 · 5 = 10

Therefore, n = 10.

We hope that the above article on Collinear Vectors is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.