Explanation of Parallel Vectors

Two vectors a and b are called parallel if the angle they form from the vertical axis or the horizontal axis (not necessarily together) is the same or equal. Another way to say it is: Two vectors a and b are called parallel if and only if the angle they form between them is zero degrees or 0°.

A pictorial representation of the parallel vector, which is in the same direction, is given in Figure I:

A pictorial representation of the parallel vector, which is in opposite directions, is given in Figure II:

A pictorial representation of non-parallel vectors, i.e., vectors that are not parallel, is given below in Figure III:

Properties of parallel vectors

• The parallel vectors are vectors that are in the same direction or exactly the opposite direction, which means if we have any vector v, which is one vector, its opposite vector will be -v. Now, these two vectors are always parallel to each other.

• Parallel vectors are sometimes known as a set of collinear vectors. Collinear vector means that the two parallel vectors are always parallel to the same line, but they may or may not be in the same direction. That means we can have more than two vectors.

• Any vector a is always parallel to itself, i.e., a is always parallel to a.

• Two vectors v and w are parallel to each other if v = bw, where ‘b’ is a scalar.

• In the above one where v = bw, (‘b’ is a scalar) v and w are in the same direction if b > 0, i.e., the scalar is positive, and both the vectors v and w are in opposite directions if b < 0, that is the scalar is negative.

The dot product of parallel vectors

The dot product of the vector is calculated by taking the product of the magnitudes of both vectors.

Let us assume two vectors, v and w, which are parallel. Then the angle between them is 0o.

Using the definition of the dot product of vectors, we have,

v.w=|v| |w| cos θ

This implies as θ=0°, we have

v.w=|v| |w| cos 0

v.w=|v| |w|

Therefore, the dot product of two parallel vectors can be determined by just taking the product of the magnitudes.

Cross product of parallel vectors 

The Cross product of the vector is always a zero vector when the vectors are parallel

Let us assume two vectors, v and w, which are parallel. Then the angle between them is 0°.

Using the definition of the cross product of vectors, we have,

v×w=|v| |w| sin

This implies as θ=0°, we have 

v×w=|v| |w| sin 0

v×w=0

Therefore, the cross-product of two parallel vectors is nothing but the zero vector.

Easy formulas to determine if a set of vectors is a parallel vector

A set of vectors are called parallel if:

1. v.w=|v| |w| or

2. v×w=0 or 

3. v=bw, where b is a scalar

Unit vector parallel to a given vector

The unit vector parallel to the given vector gives the vector in the same direction as the previous vector, but the magnitude of that unit vector is 1. That is why the vector is called a unit parallel vector.

The vector parallel to a given vector and unit in magnitude is denoted by a. The formula to find the unit vector â when the given vector is a is given by:

â= a / |a| 

Here the vector a and â are both parallel to each other as the only change in â is that the magnitude is changed by multiplying with the scalar  1/a .

So, both vectors a and â are scalar multiples of each other.

To determine the magnitude of the unit vector â, compute the module of the vector â

|â|=|a /|a||

|â|= |a/a| 

|â|=1 , which is 1.

Therefore, the vector â or a /|a| is a unit vector parallel to the given vector a, obtained by dividing the given vector by its own magnitude.

Examples of parallel vectors

1. Given two vectors, A = (4, 6) and B = (-2, -3), these two given vectors are parallel as:

The vector A can be written in scalar terms of B.

A= (4,6)

A= -2(-2,-3)

A= -2B

Hence, vector A is written in terms of the scalar “-2” to vector B. Hence, they both are parallel vectors.

2. Given two vectors, A = (-3, 7, -8) and B = (-12, 28, -32), these two given vectors are parallel as:

The vector B can be written in scalar terms of A.

B= (-12, 28, -32)

B= 4(-3, 7, -8)

B= 4A

Hence, vector B is written in terms of the scalar “4” to the vector A. Hence they both are parallel vectors.

3. Given two vectors, A = (1,3) and B = (12, 36), These two given vectors are parallel as:

The vector B can be written in scalar terms of A.

B= (12, 36)

B= 12 (1, 3)

B= 12B

Hence, vector B is written in terms of the scalar “12” to vector A. Hence, they both are parallel vectors. 

Conclusion

In conclusion, the parallel vectors are those that have the same angle from the horizontal or vertical axis or the angle between them is zero. Another way to check can be using the cross-product of the vectors. If the cross-product of two vectors is 0, they are parallel vectors. The dot product of the parallel vector can be calculated just by taking the product of the two given vectors.

In terms of parallel vectors, we do not care about them being the same in magnitude. We always worry about the direction they have. It should be either the same or exactly opposite, that is, either the angle between them should be 0o or 180o.