Collinear Vectors

Collinear vectors are considered as one of the important concepts in vector algebra. When two or more given vectors lie along the same given line, then they can be considered as collinear vectors. We can consider two parallel vectors as collinear vectors since these two vectors are pointing in exactly the same direction or opposite direction.

In this article, let's learn about collinear vectors, their definition, conditions of vector collinearity with solved examples.

Any two given vectors can be considered as collinear vectors if these vectors are parallel to the same given line. Thus, we can consider any two vectors as collinear vectors if and only if these two vectors are either along the same line or these vectors are parallel to each other. For any two vectors to be parallel to one another, the condition is that one of the vectors should be a scalar multiple of another vector.

In the above diagram, the vectors that are parallel to the same line are collinear to each other and the intersecting vectors are non-collinear vectors.

In order for any two vectors to be collinear, they need to satisfy certain conditions. Here are the important conditions of vector collinearity:

• Condition 1: Two vectors \(\overrightarrow{p}\) and \(\overrightarrow{q}\) are considered to be collinear vectors if there exists a scalar 'n' such that \(\overrightarrow{p}\) = n · \(\overrightarrow{q}\)
• Condition 2: Two vectors \(\overrightarrow{p}\) and \(\overrightarrow{q}\) are considered to be collinear vectors if and only if the ratio of their corresponding coordinates are equal. This condition is not valid if one of the components of the given vector is equal to zero.
• Condition 3: Two vectors \(\overrightarrow{p}\) and \(\overrightarrow{q}\) are considered to be collinear vectors if their cross product is equal to the zero vector. This condition can be applied only to three-dimensional or spatial problems.

Proof of Condition 3:

Let's consider two collinear vectors \(\overrightarrow{a}\) = {\(a_{x}\),\(a_{y}\),\(a_{z}\)} and \(\overrightarrow{b}\) = {n\(a_{x}\),n\(a_{y}\),n\(a_{z}\)}. We can find the cross product between them as:

\(\overrightarrow{a}\) × \(\overrightarrow{b}\) = \(\left|\begin{array}{ccc}
\boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k} \\
a_{x} & a_{y} & a_{z} \\
b_{x} & b_{y} & b_{z}
\end{array}\right|\)

= i (\(a_{y}\)\(b_{z}\) - \(a_{z}\)\(b_{y}\)) - j (\(a_{x}\)\(b_{z}\) - \(a_{z}\)\(b_{x}\)) + k (\(a_{x}\)\(b_{y}\) - \(a_{y}\)\(b_{x}\))

= i (\(a_{y}\)n\(a_{z}\) - \(a_{z}\)n\(a_{y}\)) - j (\(a_{x}\)n\(a_{z}\) - \(a_{z}\)n\(a_{x}\)) + k (\(a_{x}\)n\(a_{y}\) - \(a_{y}\)n\(a_{x}\))

= 0i + 0j + 0k = \(\overrightarrow{0}\) [Because different components of the same vector are perpendicular to each other and hence, their product is 0.]

Related Articles on Collinear Vectors

Check out the following pages related to collinear vector

• Adding Vectors Calculator
• Angle Between Two Vectors Calculator
• Handling Vectors Specified in the i-j form
• Triangle Inequality in Vectors
• Subtracting Two Vectors

Important Notes on Collinear Vectors

Here is a list of a few points that should be remembered while studying collinear vectors

• Any two given vectors can be considered as collinear vectors if these vectors are parallel to the same given line.
• Thus, we can consider any two vectors as collinear if and only if these two vectors are either along the same line or these vectors are parallel to each other.

What Are Collinear Vectors?

Any two given vectors can be considered as collinear vectors if these vectors are parallel to the same given line. Thus, we can consider any two vectors as collinear if and only if these two vectors are either along the same line or these vectors are parallel to each other. For any two vectors to be parallel to one another, the condition is that one of the vectors should be a scalar multiple of another vector.

How Do You Know if a Vector Is Collinear?

In order for any two vectors to be collinear, they need to satisfy certain conditions. Here are the important conditions of vector collinearity:

• Condition 1: Two vectors \(\overrightarrow{p}\) and \(\overrightarrow{q}\) are considered to be collinear vectors if there exists a number 'n' such that \(\overrightarrow{p}\) = n · \(\overrightarrow{q}\)
• Condition 2: Two vectors \(\overrightarrow{p}\) and \(\overrightarrow{q}\) are considered to be collinear vectors if and only if the ratio of their corresponding coordinates are equal. This condition is not valid if one of the components of the given vector is equal to zero.
• Condition 3: Two vectors \(\overrightarrow{p}\) and \(\overrightarrow{q}\) are considered to be collinear vectors if their cross product is equal to the zero vector. This condition can be applied only to three-dimensional or spatial problems.

Are Parallel and Collinear Vectors the Same?

Yes, parallel vectors and collinear vectors are the same. Two vectors are collinear vectors if they have the same direction or are parallel or anti-parallel. Two vectors are parallel if they have the same direction or are in exactly opposite directions.

How Do You Prove Three Position Vectors Are Collinear?

Consider three line segments PQ, QR and PR. If PQ + QR = PR then we can consider these three points to be collinear. The three given line segments can be translated to the respective vectors PQ, QR and PR. The magnitudes of these three vectors are equal to the length of the three line segments that are mentioned here.

Give an Example of Collinear Vectors

Consider two vectors \(\overrightarrow{P}\) = (3,4,5), \(\overrightarrow{Q}\) = (6,8,10). Two vectors are considered to be collinear if the relations of their coordinates are equal.

P₁/Q₁ = 3/6 = 1/2

P₂/Q₂​​​​​​ = 4/8 = 1/2

P₃/Q₃ = 5/10 = 1/2

Since P₁/Q₁ = P₂/Q₂ = P₃/Q₃, the vectors \(\overrightarrow{P}\) and \(\overrightarrow{Q}\) can be considered as collinear vectors.

What Are Non-Collinear Vectors?

Vectors are considered to be non-collinear when they are situated in the same plane but they are not acting along the same line of action.

How Do You Find Collinear Vectors in 3 Dimensions?

Two vectors \(\overrightarrow{P}\) and \(\overrightarrow{Q}\) are considered to be collinear vectors if their cross product is equal to the zero vector. This condition can be applied only to three-dimensional or spatial problems.