Scalar product, Physical significance, Mathematical formulation, Properties, Application, Practice Problem, FAQs.

Since our childhood we have studied “2 multiplication of 2 = 4”. As these are numbers and have magnitude only, can be multiplied easily. But what about the vectors ? Vectors have both magnitude and direction. We can multiply magnitude as done above but can we multiply direction? The Answer is yes. According to the need We can multiply vectors in two ways. First is the scalar product and second is the vector product. Here we will discuss the scalar product of vectors.

Table of content

• Scalar product
• Physical significance of Scalar product
• Dot product in component form 
• Properties of scalar vector 
• Applications of Dot product
• Practice problem
• FAQs

Scalar product

The scalar product or dot product of any two vectors and , denoted by , is defined as the product of their magnitude with the cosine of angle between them.

Where magnitude of vector

And magnitude of vector

angle between and

Physical significance of Scalar product

• The dot product between two vectors is based on the projection of one vector onto another. Specifically it tells us how much one vector is in the direction of another.
  
  

If we apply a force in some direction to an object so that the object moves, we say that work is done by the force. But the magnitude of work will be the equal to the magnitude of force in the displacement direction into magnitude of displacement. In vector form the magnitude of work is given by the dot product of the force vector and displacement vector of the object.

Dot product in component form 

Let and are the two vectors represented in component form are 

and  

where , , are along the x-axis, y-axis, and z-axis respectively

Then, the scalar product is given by

Dot product of same unit vector will be 1 because cos and of perpendicular vector will be 0 as cos . i.e. and

On putting the value we get,

The result of the dot product is a scalar. That is why the dot product is often called the scalar product. It is also called the inner product.

Properties of scalar vector 

1. is always a scalar quantity which is positive if angle between the vectors is acute (𝞱 < 900) , zero if angle between them is right angle (𝞱 = 900) and negative if angle between them is obtuse (90< 𝞱 180⁰)
   
   
2. Dot products are commutative,
3. Dot products are distributive,
4. Scalar product of two vectors will be maximum when  

1. The scalar product of a vector by itself is equal to the square of magnitude of the vector.

Applications of Dot product

1. Finding the angle between two vectors : suppose two vectors and are shown in figure

(recreate)

Then the angle between them 

2. Component of a vector on another vector (Projection)

3.Condition for orthogonality (to check whether the given vectors are orthogonal or perpendicular )

4.Calculating the work done. If the force represented by the vector and the displacement represented by the vector , then the work done

Practice problem

Q1. If , , and the angle between and is 60°, find .
Answer. , ,

.

.

Q2. If and , find
Answer. and

Then

.

Q3. Find the projection of along the vector .
Answer. Projection of along the vector is given by or ,

Where the angle between vector and vector

We know that,

Q4. If and find .
Answer.

.

.

FAQs

Q1. When the scalar product of two vectors is zero?
Answer. If the two vectors are perpendicular to each other then the scalar product will be zero.

Q2. What is the value of ?
Answer. .

Q3. What is the value of ?
Answer.

Q4. The product of a scalar with the vector is, a vector or a scalar?
Answer. Product to scalar with vectors is a vector.

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