Scalar or Dot Product

An object with magnitude and direction is called a vector represented mathematically or geometrically. A vector consists of initial points and terminal points, represented by arrow. The arrow is shown with a direction equal to that of the quantity and a length equal to the magnitude of that quantity. Vectors do not have positions, even though they have magnitude and direction. A scalar differs from a vector in that it has a magnitude but no direction. For example, acceleration, velocity, and displacement are vector quantities, while mass, time, and speed are scalars.

Scalar product 

When two vectors are multiplied together, the result you get is known as a scalar product. The scalar product of two vectors can also be described as the sum of products of their corresponding components. The answer obtained will always be a scalar quantity and, as a result, a real number. 

Having widespread application in fields like engineering and astronomy, we can also calculate the scalar product by taking the product of the magnitude of vectors alongside the cosine of the angle between them. 

We can understand this using an example: 

Let’s say we have 2 vectors,  A and B. When we multiply them, the scalar product obtained will be as follows: 

A.B=AB cos

Here is the angle between these two vectors.

Upon expansion, this can also be written as: 

A.B=AB cos = A(B cos )=B (A cos )

Here it is clear that Bcos will be the projection of  B onto A while Acos will be the projection of A onto B. Therefore, the scalar product can be defined as a product of the magnitude of A and the component of B with A or as a product of the magnitude of A and the component of B with A.

Scalar Product of Two Vectors

• The dot product is another name for the scalar product and is computed in the same way as an arithmetic operation. 
• A scalar product produces a scalar quantity, as the name implies.
• The scalar product for vector quantity cannot be calculated if the direction and magnitude are lacking.

The term “vector” refers to a quantity with both direction and magnitude. Addition and product are two mathematical operations possible on vectors. There are two methods for multiplying vectors: dot product & cross product.

Vector Cross Product and Dot Product

• The cross product of vectors or the vector product of vectors
• The only difference between the two techniques is that with the first, we get a scalar value as a result, and with the second, we get a vector value.

Applications of the Scalar Product

In vector theory, the scalar product has a variety of uses, including:

• Vector projection: The scalar product is used to establish how a vector is projected onto another vector. a.b|b| is the projection of vector a onto vector b. Similarly, a.b|a| is the projection of vector b onto vector a.
• Scalar triple product: The scalar triple product of three vectors is calculated using the scalar product. The scalar triple product’s formula is a.(b x c) = b.(c x a) = c.(a x b)
• The angle between two vectors: The formula for determining the angle between two vectors is a scalar product is cos θ = a.b|a||b|

The calculation of work is one of the applications of the scalar product. Work is defined as the dot product of the applied force vector and the displacement vector. When a force is applied at an angle to the displacement, the work is calculated as W = f d cosθ, the dot product of force and displacement. The dot product can also determine whether two vectors are orthogonal.

 a.b = |a||b| cos 90 = |a||b|=0

Properties of a Scalar Product

Now that we have grasped the concept of the scalar product, let us look at some of the critical traits of the scalar product of vectors a and b that can aid us in solving problems:

• The commutative property of a scalar product is that a.b = b.a.
• The scalar product of vectors follows the distributive property:

a.(b+c) = a.b + a.c

(a+b).c = a.c + b.c

a.(b – c)  = a.b – a.c

(a – b).c  = a.c – b.c

Because the dot product among a scalar (a.b) and a vector (c) is not defined, the phrases involved in the associative property, (a.b).c or a.(b.c), are both ill-defined. 

• Orthogonal:

If a.b = 0, two non-zero vectors a and b are orthogonal.

• There will be no cancellations:

The cancellation law does not apply to the dot product, unlike the normal product, where if ab = ac, b always equals c until a is 0.

If a.b = a.c  then it is not necessarily true that b = c .

Conclusion

The dot product, also known as the scalar product, is a significant operation performed on vectors with many use cases in mathematics and physics. From a geometric standpoint, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.