State and prove that dot product is distributive. from Physics Motion in A Plane Class 11 Manipur Board
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State and prove that dot product is distributive.

Dot product is distributive. Dot product of a given vector with a sum of number of other vectors is equal to the sum of the dot product of given vector with the other vectors separately. 

That is, straight A with rightwards harpoon with barb upwards on top. space left parenthesis B with rightwards harpoon with barb upwards on top plus C with rightwards harpoon with barb upwards on top plus... right parenthesis space equals space A with rightwards harpoon with barb upwards on top. B with rightwards harpoon with barb upwards on top space plus space A with rightwards harpoon with barb upwards on top. C with rightwards harpoon with barb upwards on top 

Proof: Let us consider that three vectors stack straight A comma with rightwards harpoon with barb upwards on top space B with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top are represented by OL with rightwards harpoon with barb upwards on top comma space stack O P with rightwards harpoon with barb upwards on top space a n d space stack O Q with rightwards harpoon with barb upwards on top spacerespectively. 

Let, the angle between straight A with rightwards harpoon with barb upwards on top space a n d space B with rightwards harpoon with barb upwards on top is straight alpha and that between straight A with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top space i s space beta

 

In the fig. above, the resultant vector is given by straight R with rightwards harpoon with barb upwards on topstraight R with rightwards harpoon with barb upwards on top is the resultant of B with rightwards harpoon with barb upwards on top space a n d space C with rightwards harpoon with barb upwards on top
Angle made by straight R with rightwards harpoon with barb upwards on top space w i t h space stack A space with rightwards harpoon with barb upwards on top i s space theta.

Construction: From P and T draw perpendiculars PM and TN on OL and draw perpendicular PS from P on TN. 

straight A with rightwards harpoon with barb upwards on top. left parenthesis B with rightwards harpoon with barb upwards on top space plus space C with rightwards harpoon with barb upwards on top right parenthesis space equals space A with rightwards harpoon with barb upwards on top. space R with rightwards harpoon with barb upwards on top

space space space space space space space space space space space space space space space space space space space space equals space A R space c o s theta space

space space space space space space space space space space space space space space space space space space space space equals space A left parenthesis O N right parenthesis space

space space space space space space space space space space space space space space space space space space space space equals space A thin space left parenthesis O M plus M N right parenthesis space

space space space space space space space space space space space space space space space space space space space space equals space A left parenthesis O M plus P S right parenthesis space

space space space space space space space space space space space space space space space space space space space space equals space A left parenthesis O P space c o s alpha space plus space P T space c o s beta right parenthesis space

space space space space space space space space space space space space space space space space space space space space equals space A thin space left parenthesis B space c o s alpha space plus space C space c o s beta right parenthesis space

space space space space space space space space space space space space space space space space space space space space equals space A B space c o s alpha space plus space A C space c o s beta

space space space space space space space space space space space space space space space space space space space space equals space A with rightwards harpoon with barb upwards on top. B with rightwards harpoon with barb upwards on top space plus space A with rightwards harpoon with barb upwards on top. C with rightwards harpoon with barb upwards on top space 

Hence, from the above result we can see that the dot product is distributive. 

                   

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