Vector Product Of Two Vectors And Their Properties

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What is a Vector

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A vector is an object with magnitude and direction. It is used to locate a point in space with respect to another point. This mathematical quantity can be used to calculate positions, displacements, velocities, and acceleration.

• It can change the proportions and order of the two vectors, but the result will remain the same in the vector space.
• The quantity is denoted by the symbol such as U, V, and W.
• Two vectors are said to be equal if they have the same direction and magnitude.
• It can be represented by a direct line segment that indicates the direction and magnitude of the quantity.
• The vector product of two vectors can be obtained by two methods, namely, dot product and cross product.

In the diagram below, we can see that the direction of the vector is from the tail to the head.

Representation of Vector

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Vector Product of Two Vectors

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A vector product of two vectors, also known as cross product or area product, is formed when two real vectors are joined by binary multiplication of two vectors in a three-dimensional space.

• If we assume the two vectors to be a and b, their vector is denoted by a x b.
• The resultant vector is perpendicular to the original vectors.
• A vector product of two vectors is demonstrated using the formula: 

a x b = |a||b|sin(θ)n

Here, 

• |a| = length or magnitude of vector a
• |b| = length or magnitude of vector b
• θ = angle between both vectors a and b
• n = unit vector perpendicular to both vectors a and b

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Magnitude of the Vector Product

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The magnitude of the vector product is determined by the area of the parallelogram formed by the initial two vectors. It is calculated using the square root of the sum of the squares of the vector components.

• Magnitude is a scalar value that cannot be negative.
• It indicates the length of the vector.
• The magnitude of the vector product of two vectors is the product of their lengths. 

Consider two vectors: A = A_x + A_y + A_z and  B = B_x + B_y + B_z

Then the magnitude of two vectors is given as:

|A| = √(A)_x²+(A)_y²+(A)_z²

|B| = √(B)_x²+(B)_y²+(B)_z²

Hence, the magnitude of the vector product of two vectors is given by the formula,

|A × B| = |A||B||sinθ|

• The fig.1 diagram below shows that θ (i.e. 360°) is the angle between the two vectors a and b.
• In the case of a vector product, we always consider the smaller angle, which is less than 180°.
• As in the fig.2 diagram below, the n represents the area perpendicular to the plane containing the two original vectors.

  

                                                                                    Fig. 1                                              Fig. 2

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Direction of the Vector Product

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The direction of the vector product of two vectors depends on the chosen orientation of the space. To understand the direction of a vector product, we can follow the convention of the right-hand screw rule. 

• Here, one takes a right-handed screw with its head lying on the plane of the vectors a and b. 
• With the screw being perpendicular to the plane surface, if one rolls its head in the direction from a to b, then the tip of the screw advances in the direction of c. 
• An easier version of this is the right-hand thumb rule.
• In this, one needs to curl up the fingers of our right hand around a line perpendicular to the plane of the vectors a and b. 
• If the fingers are curled up in the direction from a to b, then the stretched thumb should point upwards in the direction of c.
• Another way includes aligning the first finger of the right hand with a and the middle finger with b. 
• Then, one needs to position these two fingers and the thumb at right angles, with the thumb pointing in the direction of a × b. 

Direction of the Vector Product

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Properties of Vector Product

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The vector product has interesting properties, which are as follows:

Anti-Commutative Property

While the dot product result of vectors follows the commutative property, however the result is different in case of cross product. The vector product of two vectors is non-commutative. 

• Let’s consider a simple instance to understand this property. 
• The product of two vectors, a and b is indicated by a x b. 
• Now to calculate the product in a different order, we apply the formula b × a = |b| |a| sin(θ) −n.
• In this case, one sees that the direction of b × a is opposite to that of a × b ).
• Mathematically, one can conclude that b × a = −a × b, thus indicating the vector product of two vectors is not commutative.

Distributive Property

The second property of the vector product is that it is distributive over addition, just like the dot product. This means that a × (b + c) = a × b + a × c. The result remains equivalent even when the direction of the vector product is altered. Therefore, (b + c) × a = b × a + c × a. 

Zero Vector Property

According to the zero vector property, if there are two vectors, a and b, the vector product of two vectors would be 0 only if a = 0 or b = 0.