8. Slole
any
six characteristics of scaler product
Dot product
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Physics > Work, Energy and Power > The Scalar Product
Work, Energy and Power
The Scalar Product
Velocity, displacement, force and acceleration are different types of vectors. Is there a way to find the effect of one vector on another? Can vectors operate in different directions and still affect each other? The answer lies in the Scalar product. An operation that reduces two or more vectors into a Scalar quantity!
Let’s consider two vector quantities A and B. We denote them as follows:

Their scalar product is A dot B. It is defined as:
A.B = |A| |B| cosθ
Where, θ is the smaller angle between the vector A and vector B. An important reason to define it this way is that |B|cosθ is the projection of the vector B on the vector A. The projections can be understood from the following images:

Since, A(Bcosθ) = B(Acosθ), we can say that
A.B = B.A
Hence, we say that the scalar product follows the commutative law. Similarly, the scalar product also follows the distributive law:
A.(B+C) = A.B + A.C
Now, let us assume three unit vectors, i, j and k, along with the three mutually perpendicular axes X, Y and Z respectively.

As cos (0) = 1, we have:

Also since the cosine of 90 degrees is zero, we have:

These two findings will help us deduce the scalar product of two vectors in three dimensions. Now, let’s assume two vectors alongside the above three axes:


So their scalar product will be,

Hence,
A.B = AxBx + AyBy + AzBz
Similarly, A2 or A.A =
In Physics many quantities like work are represented by the scalar product of two vectors. The scalar product or the dot product is a mathematical operation that combines two vectors and results in a scalar. The magnitude of the scalar depends upon the magnitudes of the combining vectors and the inclination between them.