Vector product, right hand thumb rule, Physical significance, mathematical representation, properties, Application, Practice problem, FAQs

Do you know why the door handles are placed away from the hinges? This is because the applied force needed to open the door can be minimum. As we know the torque is the vector product of radius vector and force vector. The vector product of two vectors is equal to multiplication of magnitude of vectors with the sine of angle between them and direction is given by the right hand thumb rule. So for a constant torque if we place handles at a maximum radius and apply the force perpendicular to the door and force needed will be minimum.Figure shows the top view of a door.

Let’s learn more about the vector product.

Table of content

• Vector product
• Right hand thumb rule
• Physical significance of cross product
• Vector product of orthogonal unit vector
• Cross product when vector are given in unit vector form
• Properties of Cross product
• Application of Cross Product
• Practice problem
• FAQs

Vector product

Vector product or Cross product of two coplanar vectors is defined as the vector whose magnitude equals the multiplication of the magnitude of two vectors with the sine of angle between them and its direction is perpendicular to the plane which contains the two vectors which is given by the right hand thumb rule.

Let and are the two given vector the vector (Cross) product is given by the vector

Where magnitude of vector

And magnitude of vector

angle between and

unit normal vector perpendicular to the plane containing vector and  

Right hand thumb rule

Right hand thumb rule is used to find the direction of .The step for finding the direction is given as

1. Put the given vector and tail to tail
2. Place your right hand palm along the such that when finger is closed it should turn along as shown in figure. 
3. Now the direction of thumb is the direction of .
   
   

Physical significance of cross product

Cross product of two vectors gives the area of parallelogram whose two adjacent sides are formed by those vectors.

Consider a parallelogram having sides and as shown in figure. Then form geometry the area of parallelogram is equal to

Hence the area of the parallelogram is equal to the magnitude of the cross product of two sides of the vector.

Vector product of orthogonal unit vector

If , , are the three orthogonal unit vectors representing the three directions of cartesian coordinate i.e. x , y , z then cross product of two unit vectors can be found as,

If we put the three unit vectors on a circle as shown in figure then the cross product of any two will be equal to the third. If we move along the anticlockwise direction the product will be positive and if we move along clockwise direction the product will be negative.This circle is called vector circle diagram.

The cross product of a unit vector with itself will be equal to zero because angle between them is .

Cross product when vector are given in unit vector form

Let and are the two vectors represented in unit vector form is given by

and  

Then the cross product is given by

As we know  

and , ,

On putting the value

On rearranging we get,

This expression can be represented in determinant form as

Properties of Cross product

The Properties of vector product are given below

1. The vector product of two vectors is always perpendicular to the plane containing the vector.
2. Vector product is not commutative i.e. 

Hence,

But

3. Vector products are distributive,

4. The vector product of two vectors will be maximum when the two vectors are orthogonal.

As we know vector produce vector produce of two vector is given as

Value of sine function is maximum when , so the vector product will be maximum when vectors are orthogonal.

5. The vector product of two non-zero vectors will be minimum when they are parallel or antiparallel to each other.

As we know vector produce vector produce of two vector is given as

Value of sine function is minimum when , so the vectors will be parallel.

Application of Cross Product

1. Area of parallelogram

2. Area of triangle

3. Unit vector perpendicular to two vector

4. Moment of force

Practice problem

Q1. Given vector and find the value of
Answer: We know from the cross product 

Q2. Two vector and find the value of using determinant method

Answer: Vector product of two vector in determinant form is given as 

Q3. Calculate the area of parallelogram having adjacent side and .
Answer: Finding

Area of parallelogram

Q4. Calculate the area of triangle determined by the vectors and .
Answer: Finding

We know that the area of triangle .

FAQs

Q1. What is the magnitude of the Cross product of two perpendicular unit vectors?
Answer: Magnitude will be 1.

Q2. How many types of vector products are there?
Answer:There are two type of vector product-
A. Scalar product
B. Dot product.

Q3. Can we divide two vectors?
Answer: Two vectors cannot be divided.

Q4.The quantity which is represented by the vector product of two vectors is a scalar quantity or vector?
Answer:The quantity is a vector.