Parallel Vectors

The parallel vectors are vectors that have the same direction or exactly the opposite direction. i.e., for any vector a, the vector itself and its opposite vector -a are vectors that are always parallel to a. Extending this further, any scalar multiple of a is parallel to a. i.e., a vector a and ka are always parallel vectors where 'k' is a scalar (real number).

Let us learn more about parallel vectors along with its definition, formula, and examples.

Two vectors are said to be parallel if and only if the angle between them is 0 degrees. Parallel vectors are also known as collinear vectors. i.e., two parallel vectors will be always parallel to the same line but they can be either in the same direction or in the exact opposite direction. In the following image, the vectors shown in the left-most figure are NOT parallel as they have different directions (i.e., neither the same nor opposite directions).

The parallel vectors that are in opposite directions are sometimes referred to as anti-parallel vectors too. In the above image, the last figure shows the anti-parallel vectors. But how to identify the parallel vectors mathematically? Let's see.

Two vectors a and b are said to be parallel vectors if one is a scalar multiple of the other. i.e., a = k b, where 'k' is a scalar (real number). Here, 'k' can be positive, negative, or 0. In this case,

• a and b have the same directions if k is positive.
• a and b have opposite directions if k is negative.

Here are some examples of parallel vectors:

1. a and 3a are parallel and they are in the same directions as 3 > 0.
2. v and (-1/2) v are parallel and they are in the same directions as (-1/2) < 0.
3. a = <1, -3> and b = <3, -9> are parallel as b = <3, -9> = 3 <1, -3> = 3a.

In the above examples, example 2 refers to the anti-parallel vectors.

The dot product of any two parallel vectors is just the product of their magnitudes. Let us consider two parallel vectors a and b. Then the angle between them is θ = 0. By the definition of dot product,

a · b = |a| |b| cos θ
= |a| |b| cos 0
= |a| |b| (1) (because cos 0 = 1)
= |a| |b|

Therefore, the dot product of two parallel vectors is the product of their magnitudes.

The cross product of any two parallel vectors is a zero vector. Consider two parallel vectors a and b. Then the angle between them is θ = 0. By the definition of cross product,

a × b = |a| |b| sin θ \(\hat{n}\)
= |a| |b| sin 0 \(\hat{n}\)
= |a| |b| (0) \(\hat{n}\) (because sin 0 = 0)
= 0

Note that 0 here is a vector, not a scalar. Thus, the cross-product of two parallel vectors is a zero vector (not simply zero).

The parallel vectors can be determined by using the scalar multiple, dot product, or cross product. Here is the parallel vectors formula according to its meaning explained in the previous sections.

The unit vector that is parallel to a given vector a is denoted by \(\hat{a}\) and is given by \(\hat{a}\) = a / |a|. Observe two things here:

• a and a / |a| (which is 1/|a| · a) are scalar multiples of each other. Hence, a and \(\hat{a}\) are parallel.
• The magnitude of a / |a| is |a| / |a| = 1. Hence \(\hat{a}\) is a unit vector.

Hence, a / |a| is a unit vector parallel to a. It is obtained by dividing a vector by its magnitude.

Example: Find the unit vector that is parallel to the vector a = 3i + 4j.

Solution:

It is given that a = 3i + 4j.

Its magnitude is |a| = √(3² + 4²) = √(25) = 5.

Thus, the unit vector parallel to a is,

\(\hat{a}\) = a / |a|
= (3i + 4j) / 5
= (3/5)i + (4/5)j

• Two vectors a and b are parallel to each other if and only if a = kb, where 'k' is a scalar.
• Here, a and b are in the directions if k > 0 and are in opposite directions if k < 0.
• Every vector a is parallel to itself as a = 1 a.
• Two vectors a and b are said to be parallel if their cross product is a zero vector. i.e., a × b = 0.
• For any two parallel vectors a and b, their dot product is equal to the product of their magnitudes. i.e., a · b = |a| |b|.

☛ Related Topics:

• Vector Formulas
• Components of a Vector
• Types of Vectors
• Resultant Vector Calculator

What is Parallel Vectors Definition?

Two vectors a and b are said to be parallel vectors if one of the conditions is satisfied:

• If one vector is a scalar multiple of the other. i.e., a = kb, where 'k' is a scalar.
• If their cross product is 0. i.e., a × b = 0.
• If their dot product is equal to the product of their magnitudes. i.e., a · b = |a| |b|.

How Do You Find a Vector Parallel to a Given Vector?

To find a vector that is parallel to a given vector a, just multiply it by any scalar. For example, 3a, -0.5a, √2 a, etc are parallel to the vector a.

How Can You Determine if Two Vectors are Parallel?

To determine if two given vectors are parallel, just see whether you can take a common factor out of one vector so that it is a multiple of the other vector. Another way is to check whether their cross product is 0.

What is the Difference Between Perpendicular and Parallel Vectors?

Here are the differences between perpendicular and parallel vectors.

Is a Vector Parallel to Itself?

Every vector a is a scalar multiple of itself. i.e., a = 1a. So every vector is parallel to itself. Also, the angle between a vector and itself is always 0 degrees. In this way also we can tell that a vector is parallel to itself.

What is the Formula for Unit Vector Parallel to the Resultant Vectors?

We know that the unit vector parallel to a vector a is a / |a|. So the unit vector parallel to the resultant of two vectors a and b is (a+b) / |a+b|.

What Is the Difference Between Parallel Vectors And Skew Lines?

Parallel vectors and the skew lines are both in the three-dimensional space. The parallel lines never intersect and are parallel with reference to the x, y, and z coordinates. The skew lines are also in the three-dimensional space, but are neither parallel nor are them intersecting. The skew lines are the line present in different planes.

What are Equal and Parallel Vectors?

Equal vectors have the same magnitude and same direction. The parallel vectors may have different magnitudes but they have the same/opposite directions. For example:

• a and a are equal vectors.
• a and 3a are parallel vectors.