Vector Product of Two Vectors

A vector is used to locate a point in space concerning another and is a quantity having magnitude and direction. In a geometrical representation, it can be pictured as an arrow or a line segment with the direction. Here, the length of an arrow indicates the magnitude of this vector. 

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When multiplying two vectors, it can be done using two methods. 

  1. Dot product or Scalar product

  2. Cross product or Vector product 

Here, we will discuss the cross product in detail. 

Vector Product of Two Vectors

The cross product or vector product obtained from two vectors in a three-dimensional space is treated as a binary operation and is denoted by x. The resulting product, in this case, is always another vector having some magnitude and direction. 

Let us consider the two quantities vector a and b, as shown in this image below. 

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Two vectors a and b are shown in the picture. To answer what a vector product, look at the calculations below. 

a x b = |a| . |b| . Sin(Ꝋ) n 

Where, 

  • |a| is the length or magnitude of vector a. 

  • |b| is length or magnitude of vector b. 

  • Ꝋ is angle between both the vectors b and a. 

  • n is a unit vector perpendicular to both vectors a and b. 

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As shown in the above picture, if the tail of vectors b and a begins from the origin (0,0,0), then the product of two vectors can be represented as 

Cx = ay . bz – az . by

Cy = az . bx – ax . bz

Cz = ax . by – ay . bx

Example 

Let us define a vector product by taking an example. Consider a vector a = (2,3,4) and b = (5,6,7). Here, ax = 2, ay = 3, and az = 4. bx = 5, by = 6, bz = 7. Putting these values in the above equation and calculating, we get Cx = -3, Cy = 6, and Cz = -3. 

Direction of Product Vector 

While you can define a vector product of two vectors and its magnitude from the above equation, the direction of its product vector can be determined using the rule of right-hand thumb. 

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According to this right-hand thumb rule, we need to curl the fingers of your right hand from vector a to vector b, and then the thumb is pointed towards the direction of the product vector. 

Properties of Vector Product

  • Commutative Property 

While the scalar or dot product result of two vectors shows the commutative property, and the cross product is non-commutative in nature. 

This means, a x b ≠ b x a. However, from the definition of vector product, a x b = - b x a. This is true because of the change in direction of the product vector. 

  • Distributive Property

Similar to a scalar product, this vector product determined from multiplying two vectors also shows a distributive property. 

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

As per the characteristics of the vector product, this calculation of magnitude value of the vector product equals the area of parallelogram made by the same two vectors. 

You will be able to understand the concepts of vector and cross products intricately by going through our study materials. Now, you can also download our Vedantu app for easier access to these study material, along with the option of online interactive classes to clear your doubts. 

FAQ (Frequently Asked Questions)

1. Is a x b Equal to b x a, where they are Two Vectors?

Ans. The magnitude of a x b and b x a is the same; however, their direction differs. So, effectively a x b = - b x a. 

2. Write down the Formula for a Cross Product or Vector Product. 

Ans. The formula determining the cross product of two vectors is a x b = |a| . |b| . Sin(Ꝋ) n, where |b| and |a| are values of vector b and a respectively and Ꝋ is the angle formed between them. 

3. Which Rule is Used to Determine the Direction of a Vector Product?

Ans. The right-hand thumb rule states, when you curl your right-hand fingers from the direction of vector a to vector b, then your thumb indicates the direction of the resulting vector.