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Question 5: Define the dot product (scalar product) of two vectors. What geometric interpretation does the dot product have? Give examples.

Answer

Definition of scalar product

Scalar or dot product of two vectors is defined as the product that yields a result that is a scalar quantity. Mathematically,

Where θ is the smaller angle between the vectors.

Explanation

Let A and B are two vectors and θ is the smaller angle between them when placed tail-to-tail. Then the scalar product of the two vectors is;

Here A is the magnitude of A and B is the magnitude of B. They are both numbers. Similarly, cosθ is also a number. Therefore, the product is simply the multiplication of three real numbers and a scalar.

Scalar product is commutative

As the product of real numbers is commutative (commutation means the order is irrelevant and we can multiply them in any order), therefore, the scalar product is also commutative.

Geometric interpretation of scalar product

Geometrically, scalar product is the product of magnitude of vector A with the projection of vector B on A.

Consider the figure to the right. A and B are two vectors and θ is the smaller angle between them when placed tail to tail. Bcosθ is the orthogonal (i-e, perpendicular) projection of B on A. 

This means the scalar product of A and B is A times the projection of B on A.

Since A.B = B.A = B(Acosθ). Therefore, we can equally say that it is B times the projection of A on B.

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