As it is popularly called, the cross product or vector product is one of the influential calculations on Euclidean vector space. The cross product is not the same as the dot product as it is only useful for the determination of 3-dimensional oriented Euclidean vector space. This is unlike the Dot product that can be used in any vector space, despite certain limitations in calculations.

Generally, the Vector product is a binary operation on two distinct vectors in a 3-dimensional oriented Euclidean vector space. The binary operation is denoted by X, while the Euclidean vector space is denoted by *E*.

For the cross product calculation to be possible between two vectors, certain basic conditions must be met. The first is the need for the two vectors to be linearly independent. The two vectors are usually the only vectors in the Euclidean space until the cross operation is applied; a X b = a b = c. The X is called a ‘cross’ while c = ab is the new vector formed from the X operation of the two vectors. C, when formed, must be perpendicular to the two vectors a and b, making it normal to their plane.

So, the Vector product is simply nothing more than the ability to identify a new product while keeping to the above rules. Of course, it is quite more than that, as certain properties must be kept to determine if finding the new vector is feasible.

This vector operation is extremely important across different fields and is well employed by experts in breaking down complex calculations. It is relevant in Mathematics, geometry, Physics, and Engineering. This article will outline everything that you need to know about vector products and how to caudate it.

**Cross Product Formula**

The cross product of two vectors a X b is defined as a new vector c perpendicular to a and b (the original vectors), with a direction determined by the right-hand rule and a magnitude equal to the area spanning both initial vectors.

**By definition, the formula is;**

a X b = ‖a‖‖b‖ )n

where

is the degree of the area between the two vectors a and b in the Euclidean plane

‖a‖is the magnitude of a

‖b‖is the magnitude of b

n is the unit vector to the plane of the vectors of a and b

**An alternative form of the formula is;**

a** **** **X b** **= c

Where;

a = ax ay az is the first linear independent vector

b = bx by bz is the second linear independent vector

c = cx cy cz is the cross product formed from the operation of the two vectors

This formula is based on a matrix, and the enclosed variables define the magnitudes of the respective vectors. This alternative vector is the most compatible method for calculation as it employs the matrix and will be the preferred option for calculation in this text.

**How to Calculate the Vector Product?**

**For the first formula**

Determine the degree of the area between a and b and compute

Determine the magnitude of the vectors a and b

Determine the unit vector of the direction, n

Compute to find C

**When the degree of the area is not readily available, the second formula becomes the viable option**

**Steps to calculate the second formula**

Manipulate the vectors a and b into a matrix

Find the determinant of the matrix

The answer reached is the new vector C

**Properties of the Vector Products**

The graph below shows the two vectors a and b in a 3-dimensional space with their vector product as c = a x b.

The popular right-hand method is a great way to determine the 3-dimensional Euclidean method when calculating the cross product of vectors.

The hand expresses the same description as the paragraph above, with the two last curved fingers showing exactly the direction of the vectors.

Both diagrams are to enhance the understanding of the operation in a three Euclidean plane. The properties of the cross-product will now be considered before practical examples are established.

**The Properties**

The vector C must be perpendicular to vectors a and b, which means that it must intersect both vectors on the Euclidean plane such that it forms an angle with the two

If the vectors a and b are moving in the same direction or in the full opposite direction, then their a X b will be 0. That is because they moving in the same direction signifies that an area degree does not exist i.e = 0, while both lines moving in the opposite direction means that they are parallel and will make their degree = 0.

Now Sin 0^{0} = Sin 180^{0 }= 0

Placing that in the formula will be

a X b = ‖a‖‖b‖ )n = a X b = ‖a‖‖b‖ x 0 x n = 0

diagrammatically, this is what this property means

For the second case;

The two vectors will not lead to a new vector ‘c’ if the above rules are not maintained. For example, there is the need for the two vectors to be linearly independent for a new one to be formed, so if the two vector

The cross product is not commutative. i.e a X b = b X a. Instead a X b = – a X b

It is distributive i.e a X (b + c) = a X b + a X c

Since the C is perpendicular to a and b, it means that a . c = b . c

**Examples **

Calculate the cross product of the following vectors

a = 3i + 5j – 7k

b = 2i – 6l + 4k

**solution. **

All that needs to be done is to implement the second.

**Step 1: ****Manipulate the vectors ****a ****and b into a determinant matrix**

To do this, create a 3×3 matrix with the first row having the common variables in the two vectors i.e I, j, k

i j k

Now input all the components of the first vector

i j k 3 5 -7

Do the same with the components of the second vector

i j k 3 5 -7 2 -6 4

Therefore

a X b = i j k 3 5 -7 2 -6 4

**Step 2: Find the determinant of the matrix**

To Find the determinant of a 3 x 3 matrix, you should know that every single component in the first row are assigned signs. That is

+ – + which means that the expression of the values when finding a 3 x 3 determinant would be

+i -j +k 3 5 7 2 -6 4

The next step is to manipulate all the components in the first row such that they are multiplying the determinant of a secondary 2 x 2 matrix formed in the process.

For example, the first component in the first row is i, so you must not consider any elements sharing a column or row with it such that the remaining values in the 3 x 3 matrix form a 2 x 2 matrix whose determinant will multiply i.

i.e; i j k 3 5 -7 2 -6 4 = +i 5 -7 -6 4 = i x 5 -7 -6 4

The next number is j so it will take a similar process

i.e i j k 3 5 -7 2 -6 4 = -j 3 -7 2 4 = – j x 3 -7 2 4

for k, the calculation will be

i j + k 3 5 -7 2 -6 4 = k 3 5 2 -6 = k x 3 5 2 -6

So a X b i j k 3 5 -7 2 -6 4 = i x 5 -7 -6 4 – j x 3 -7 2 4 + k x 3 5 2 -6

At this stage, the determinant of the 2 x 2 matrices must be determined. The formula for the determinant of a 2 x 2 matrix a b c d = ad – bc

Therefore

a X b = i x (20 – 42) -j x (12 – – 14) k x (-18 – 10)

a X b = -22i -26j -28k

**Step 3: The answer reached is the new vector C**

Therefore C = a X b = -22i – 26j – 28k

And the vector C = -22 26 28

To know if c is indeed the cross product of a and b, the last property that states a . c = b . c if C is perpendicular to a and b will be employed

For a . c;

a . c = a_{x}c_{x} + a_{y}c_{y } + a_{z}c_{z}

a . c = 3(-22) + 5(-26) + (-7)(-28)

a . c = -66 – 130 + 196

a . c = 0

For b . c;

b . c = b_{x}c_{x} + b_{y}c_{y } + b_{z}c_{z}

b . c = 2(-22) + (-6)(-26) + 4(-28)

b . c = -44 + 156 – 112

b . c = 0

From the above caculations; a . c = b . c = 0.

**Example 2**

a = 5 -4 3

b = -7 2 -8

Find the cross product of the two vectors a x b = c

**Solution**

The vectors will need to be arraned as an equation

a = 5i + -4j + 3k

b = -7i + 2j – 8k

a X b = i j k 5 -4 3 -7 2 -8

Following the same process as the first example, you will arrive at

a X b = i x -4 3 2 -8 – j x 5 3 -7 -8 + k x 5 -4 -7 2

a X b = i(32 -6) – j(-40 + 21) + k(10 -28)

a X b = 26i + 19j -18k

c = a X b = 26 19 -18

To check if C is true compare whether a . c = b . c

a . c = a_{x}c_{x} + a_{y}c_{y } + a_{z}c_{z}

a . c = 5(26) + -4(19) + 3(-18)

a . c = 130 – 76 – 54

a . c = 0

For b.c;

b . c = b_{x}c_{x} + b_{y}c_{y } + b_{z}c_{z}

b . c = -7(26) + 2(19) + -8(-18)

b . c = -182 + 38 + 144

b . c = 0

From the above caculations; a . c = b . c = 0.

Any example you will encounter on vector product will look similar to the above two examples. You need to ensure that you are very sound with 2 x 2 and 3 x 3 matrix determinants as they play a major role in computing a vector.

You should also note that a new vector cannot be a scalar via this operation, shown in the two examples above.

**Conclusion**

Calculating the vector product of two vectors requires a good level of Euclidean spaces to understand the essence of the calculation. The article specifies that this operation is only relevant to a 3-Euclidean space. This is unlike the Dot product, which can be used to determine different vector computations. Also, unlike the Dot product, the answer for this binary operation is a vector and not a scalar.

The vector product has proven to be very important in determining another vector from two linearly independent, and this possibility has led to the employment of the operation in different fields such as mathematics, physics, geometry, and Engineering.

Another important point to note in this article is the methods of computing the operation. There are two formulas, with the first one being mathematical calculation. However, determining the magnitude and direction of a new vector through the first formula may be difficult, so using the alternative formula is considered a more realistic solution.

Generally, the second formula explores the basic understanding of 2×2 and 3×3 matrices. The major matrix calculation is the determinant, which is very easy to determine from its formula. Also, as expressed in the last property of the operation and practical examples, if the dot product of the new vector and one of the initial vectors is equal to the dot product of the new vector and the other vector, then the new vector components is correct. That is a . c = b . c.

The other properties expressed with diagrammatic explanations also emphasize the strength and weakness of the operation and give a very good idea of when it should be employed. So basically, all you have to do to always get the right computation of this operation is to follow the outlined properties and how the established method of computation.