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Some physical quantities are themselves a scalar quantity, but are composed from the product of vector quantities. One such example is “work”. On the other hand, there are physical quantities like torque and magnetic force on a moving charge, which are themselves vectors and are also composed from vector quantities.
Thus, products of vectors are defined in two distinct manner – one resulting in a scalar quantity and the other resulting in a vector quantity. The product that results in scalar value is scalar product, also known as dot product as a "dot" ( . ) is the symbol of operator for this product. On the other hand, the product that results in vector value is vector product, also known as cross product as a "cross" ( x ) is the symbol of operator for this product. We shall discuss scalar product only in this module. We shall cover vector product in a separate module.
Scalar product of two vectors a and b is a scalar quantity defined as :
where “a” and “b” are the magnitudes of two vectors and “θ” is the angle between the direction of two vectors. It is important to note that vectors have two angles θ and 2π - θ. We can use either of them as cosine of both “θ” and “2π - θ” are same. However, it is suggested to use the smaller of the enclosed angles to be consistent with cross product in which it is required to use the smaller of the enclosed angles. This approach will maintain consistency with regard to enclosed angle in two types of vector multiplications.
The notation “ a.b ” is important and should be mentally noted to represent a scalar quantity – even though it involves bold faced vectors. It should be noted that the quantity on the right hand side of the equation is a scalar.
The angle between vectors is measured with precaution. The direction of vectors may sometimes be misleading. The basic consideration is that it is the angle between vectors at the common point of intersection. This intersection point, however, should be the common tail of vectors. If required, we may be required to shift the vector parallel to it or along its line of action to obtain common point at which tails of vectors meet.
See the steps shown in the figure. First, we need to shift one of two vectors say, a so that it touches the tail of vector b . Second, we move vector a along its line of action till tails of two vectors meet at the common point. Finally, we measure the angle θ such that 0≤ θ≤π.
We can read the definition of scalar product in either of the following manners :
Recall that “bcos θ” is the scalar component of vector b along the direction of vector a and “a cos θ” is the scalar component of vector a along the direction of vector b . Thus, we may consider the scalar product of vectors a and b as the product of the magnitude of one vector and the scalar component of other vector along the first vector.
The figure below shows drawing of scalar components. The scalar component of vector in figure (i) is obtained by drawing perpendicular from the tip of the vector, b , on the direction of vector, a . Similarly, the scalar component of vector in figure (ii) is obtained by drawing perpendicular from the tip of the vector, a , on the direction of vector, b .
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