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Explainthe geometricalinterpretation of scalar product of two vectors . |
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Answer» SOLUTION :Suppose the scalarproduct of two vectros `vec(A) and vec(B)` is given as in the figure (a) `vec(A) . vec(B) = | hat(A) ||hat(B)| cos theta ` ` vec(A) . vec(B) = AB cos theta""` …(1) which is scalar .  where`theta `is the angle between `vec(A) and vec(B)` This product is shown in two ways. Method 1 : According to figure (b) , drawa perpendicular from the head of `vec(B)` on `vec(A)` resultingOM which is shownas in figureOM is the projection of`vec(B)` on to `vec(A)` or it is called the component of `vec(B )` inthe direction of `vec(A)` .  ` :.` O = component of `vec(B) ` along `vec(A)` ` = B cos theta ` ` :. vec(A) . vec(B ) = ABcos theta ` ` = A( B cos theta )` =A(OM) = Magnitude of `vec(A ) xx` component of `vec(A) ` along `vec(A)` Method 2 : According to figure (C ) , draw a perpendicular from the head of `vec(A)` on `vec(B)` resulting ON which is shownas in figure ON is the projection of `vec(A)` on a `vec(B )`or it is called component of `vec(A)` in the direction of `vec(B)`  ` :. ` ON = component of `vec(A )` along `vec(B)` ` = Acos theta ` ` :.vec(A).vec(B) = AB cos theta ` `= B(Acos theta)` = B(ON) = MAGNITUDEOF `vec(B) xx ` component of `vec(A) ` along `vec(B)` Hence , the scalar product of two vectors is equal to theproductof the magnitudeof first vectorwith theprojectionof secondvectorthe firstvector . |
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