Calculate Vector Product

Vector Product

A vector has both the direction which is indicated by an arrow as well as the magnitude which is indicated by the length. A vector product is a combination of two vectors i.e., scalar and vector. Therefore, we have two ways in which we can multiply the vectors. First is the dot product of vectors which is also known as the Scalar product. Another is the cross product of vectors which is also known as the vector product. By the end of this, we will surely be able to define and calculate the vector product when the two vectors are provided in a cartesian form and learn the geographical applications of it. 

Define Vector Product

Now, can you define a vector product? You can start with an example. Here are two vectors (a and b) and the angle between them is represented as

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We, by now, know that when two vectors are multiplied, the result is always a vector. So, to obtain a vector, we will first have to specify the direction. And by the definition, the direction of the vector product is at the right angles to both a and b. This also means that they are at right angles even to the plane in which a and b lies. 

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Therefore, we have two choices. To make this choice we can draw help from the right-hand screw rule. According to this rule, the direction of the vector product would be in the same direction as the direction in which the screwdriver would turn, i.e., from a to b.   

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The vector product of a and b is to be defined as: a x b = |a||b| sin \[\theta\] \[\widehat{n}\]

Where, |a| is the modulus or the magnitude of a, 

             |b| is the modulus of b 

               \[\theta\] is the angle between a and b

           \[\widehat{n}\] is the unit of vectors which is perpendicular to both a and b. 

Note: Vector product is also called cross vector product as the symbol of vector product is x

Properties Of Vector Product

Before proceeding forward, there are few properties of vector products that we must know. These are:

  1. The order in which we perform the calculation matters, as a x b and b x a, are opposite to each other. Therefore, the vector product is not commutative. 

  2. The vector product is always distributive over addition, for example:

                      a x (b + c) = a x b + a x c

These are the basic vector product properties that will be helpful for you. 

Cross Vector Product Of Two Parallel Vectors

Consider the two vectors (a and b) parallel but the definition of vector does not apply to parallel lines as two parallel vectors do not define a plane. Therefore, the vector product of the two parallel vectors will be zero. 

Cross Vector Product Of Two Parallel Vectors In Cartesian Form

We can find the vector product of two vectors in a Cartesian form such as a = 3i - 2j + 7k and b = -5i +4j - 3k, where i, j, and k are the unit vectors in the directions of the x, y and z axes respectively. We can use a formula which we will develop in the end. So, first, let us start with a few cross-product examples: 

Example 1) consider that we want to find i x j. Now since they lie along the x and y axes, we can say that these vectors are perpendicular. 

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Here, we can see that k is the unit vector perpendicular to i and j and the angle between i and j is 90 degree and sin 90 degree is 1. With the help of hand screw rule, we can find i x j. Therefore,  i x j = |i||j| sin 900 k

                         = (1) (1) (1) k

                         = k

Example 2) Now if we find j x i using the hand screw rule, the vector perpendicular to j and i is equal to -k. Therefore, j x i = -k

Example 3) Finding i x i will result in zero as they are perpendicular and the angle between them is 00. therefore, i x i = 0

Based on these cross product example, we can summarize the following as:

  1. i x i = 0

  2. j x j = 0

  3. K x k = 0

  4. i x j = k

  5. j x k = i

  6. k x i = j

  7. j x i = -k

  8. k x j = -i

  9. i x k = -j 

The following cross-product example can be used to form the formula for finding the vector product of two vectors in cartesian form. 

a= a1i + a2j + a3k and b=b1i + b2j + b3k then,

 a x b = (a1i+a2j+a3k) x (b1i+b2j+b3k)        

          = a1i x (b1i+b2j+b3k) + a2j x (b1i+b2j+b3k) + a3k x (b1i+b2j+b3k)         

         = a1i x b1i + a1i x b2j + a1i x b3k + a2j x b1i + a2j x b2j + a2j x b3k +a3k x b1i + a3k x b2j + a3k x b3k       

        = a1b1i x i + a1b2i x j + a1b3i x k + a2b1j x i + a2b2j x j + a2b3j x k + a3b1k x i + a3b2k x j + a3b3k x k

Now, according to the summarization we did above, three of these terms are zero. Therefore, 

a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k is the cross product of two vectors formula that we can use to calculate a vector product in cartesian components of two vectors.

FAQ (Frequently Asked Questions)

1. Is the Dot Product and the Cross Product of a Vector Same?

No, they are not the same because dot product results in a number and cross-product result in a vector. We can take an example to understand it. Consider two arrows a and b. If we ask the question how many arrows can “a” resolve along with “b”, the answer of dot product would be that arrow a resolves just the same amount of arrows as arrow b. This would be a dot product of vector.  But according to the cross product, it asks what if the arrow sticks out of the plane which is formed by the rotation of arrow a towards b? The answer would be that if by rotation arrow a towards b, a new arrow c points up then if the arrow is rotated from b to a, the new arrow c would point down. Is the answer to both the products the same? No, it is not since the dot product is a number, and the cross product is a vector.