Vector Product of Two Vectors

Introduction

Before talking about the Vector Product Of Two Vectors, it is essential to start from the basics. Do you know what is the one thing that is common among physicists, video game creators, and air-traffic controllers? Yes, Vector! Now, what is that? And what is its significance in day-to-day life?

Well, you need to understand what scalar is before knowing it. A scalar quantity has magnitude. You know how much of something is there because of a scalar. For example, the volume of tea in your cup. Now, let’s take a look at the vectors.

What is a Vector?

Vector was founded in the 19th century by J. Willard Gibbs and Oliver Heaviside. It helped various physicists to calculate direction and speed from a graph.

Vector quantities also have a magnitude and direction. Let’s take up an instance. You need to understand how far a chair is from you to move towards it. Here, not only distance matters but displacement too. See the following diagram to have a better idea.

Source: https://www.youtube.com/watch?v=ml4NSzCQobk&t=107s

Now, vectors are unique because they remain constant. They don’t change and remain invariable with the coordinates.

Look at this example. You and your friend are in a straight line. You took three steps forward towards your friend, and he took three steps back in the same line.

Now you took two steps right, and he took two steps left. Was there any change? No. Because you both are walking in the same parallel plane. This is all because of vectors.

What is the Vector Product of Two Vectors?

Vector Product or Cross Product of two Vectors is the multiplication method of two vectors. Generally, a vector product is denoted by a multiplication sign.

When we multiply two vectors, the resultant cross-product is perpendicular to both vectors. The cross-product or vector product of the two vectors is generally a binary vector operation. Also, it is defined in three-dimensions.

The area of a parallelogram between them defines its magnitude. Whereas its direction can be found by the right-hand thumb rule. We will discuss this later in this section. Now, what is it, scalar or vector in nature? Well, the vector product of the two vectors will be vector.

Cross-Product Formula

You can find the area between two vectors by this cross-product formula.

Let there be two vectors a = a1 i + a2 j + a3  k

And

b = b1 i + b2 j + b3 k

Let y be the angle formed between a and b, where m is the unit vector.

Note that the unit vector is perpendicular to both a and b.

Hence, the cross-product will be given by

a ⃗    b ⃗ = |a| |b| sin y m ̂

Here, the vector a magnitude=  |a|

And the vector b magnitude = |b|

Properties of Cross Product

The cross product of the Vector Product Of Two Vectors has numerous properties. These properties help us understand vector multiplication. This, in turn, helps to perform various calculations. Some of these properties are

• The length of the Vector Product Of Two Vectors is a ⃗    b ⃗ = |a| |b| sin y m ̂
• Anti-commutative property is represented as a   b= -b   a
• Distributive property is given by: a (b+ c) = (a   b)+( a c)
• The Cross product of the vector 0 is given by a 0 = 0
• Vector product of the same vector: a a = 0
• Resultant of the multiplication with a scalar quantity:   x y z=xyz=y  (x z)
• Unit vector cross products: a a= b b= c c = 0

The Vector Product of a vector with the Vector Product Of Two Vectors is known as a Triple Cross Product. It, again, is a vector.

Right-Hand Thumb Rule

The Right-Hand Thumb Rule helps us find the vector direction which was produced as a result of the Vector Product Of Two Vectors. You need to take an example for this.

Suppose there is a vector a and another vector b. Then there is another cross-product c. Make sure to follow the following procedures to find the direction through this rule:

• Rotate your fingers on the right end toward vector B.
• Now, you can see that your thumb will tell you the direction for the vector c.
• In other words, the direction of the vector c would be upwards where your thumb is.
• We can now see that the resultant vector c is perpendicular to vectors a and b.

Look at the following diagram to understand better.

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