Scalar Product

Most of the quantities we are familiar with may be classified either as a scalar or a vector variable. The distinction between scalar and vector quantities is significant. Scalar quantities have only a magnitude and no direction, and their outcomes are easily calculable. However, both magnitude and direction must be available for vector quantities, and therefore, the computed outcome will also be based on the direction. Displacement, torque, momentum, acceleration, velocity and force are some examples of vector quantity.

There are two forms of vector products that can calculate the consequence of vector quantities. The first is true scalar multiplication which produces a scalar product, and the second is vector multiplication, which produces just a vector product.

What exactly is a scalar product?

The scalar product comprises two or more vectors’ equivalent components. As the names imply, a scalar product produces a scalar quantity or a real number. The product of the magnitude of the vectors and the cosine of the angle between them can also be used to determine the scalar product. Let us see the scalar product formula for 2 and 3 vectors.

Scalar product of two vectors

Dot product is another name for 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 multiplication are two mathematical operations that can be done 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.

Scalar product formula of the two vectors

The product of the modulus of the first and second vectors and the cosine of the angle among them yields the scalar product of two vectors. In other words, the scalar product is the sum of the first vector’s magnitude and the projection of the first vector onto the second vector. For two vectors a and b, the scalar product formula is

a.b = |a| |b| cosθ

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)

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 done is one of the applications of the scalar product. Work is defined as the  dot product of applied force vector and the displacement vector. When a force is applied at an angle to the displacement, the work done is calculated as W = f d cos, 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 key 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 normal multiplication, where if ab = ac, b always equals c until a is 0.

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

Conclusion:

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