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The terms 'work', 'energy' and 'power' are frequently used in everyday language. A farmer ploughing the field, a construction worker carrying bricks, a student studying for a competitive examination, an artist painting a beautiful landscape, all are said to be working. The capacity to do work is energy. We admire a long distance runner for her stamina or energy. The word 'power' is used in everyday life with different shades of meaning. In karate or boxing we talk of 'powerful' punches. These are delivered at a great speed. The aim of this chapter is to develop an understanding of these three physical quantities. Before we proceed to this task, we need to develop a mathematical prerequisite, namely the scalar product of two vectors.

In physics, work is said to be done, if a force acting on a body is able to actually move it through some distance in the direction of the force. The watch-man of the office gate is not making any effort to move but is simply standing there i.e. both the force and displacement are zero and likewise no work is done by him. Again, when the coolie carries load on his head, he exerts force along the vertical direction to support the load on his head. Since distance is covered along the horizontal i.e.. no distance is covered in the direction of the force supplied along vertical, the work performed by the coolie is also zero.

The Scalar Product
Physical quantities like displacement, velocity, acceleration, force, etc., are vectors. A scalar product gives a scalar from two vectors and a vector product produces a new vector from two vectors. Here we take up the scalar product of two vectors. The scalar product or dot product of any two vectors A and B, denoted as A.B (read A dot B) is defined as  

Where 'θ ' is the angle between the two vectors. Each vector A and B has a direction but their scalar product does not have a direction, so

Geometrically, B cos θ is the projection of B onto A and A cos θ is the projection of A onto B. So, A. B is the product of the magnitude of A and the component of B along A. Alternatively, it is the product of the magnitude of B and the component of A along B.

The scalar product follows the commutative law: A. B = B. A

And also the scalar product obeys the distributive law: A. (B + C) = A. B + A. C
Further, A· (λ B) = λ (A. B)
where λ is a real number.
The proofs of the above equations can be discussed as follows.
For unit vectors i, j, k we have
i. i = j. j = k. k =1
i. j = j. k = k. i =0
Given two vectors
A = Axi + Ayj + Azk
B = Bxi + Byj + Bzk  their scalar product is
A. B = (Axi + Ayj + Azk) . (Bxi + Byj + Bzk)
      = AxBx + AyBy + AzBz
From the definition of scalar product,
we have,
A. A = AxAx + AyAy + AzAz
A2 = Ax2 + Ay2 + Az2
Since A.A = |A ||A| cos 0 = A2
(ii) A.B = 0, if A and B are perpendicular.

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