Show that: from Physics Motion in A Plane Class 11 Manipur Board
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Motion in A Plane

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Physics Part I

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Physics

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open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar less-than or slanted equal to open vertical bar straight A with rightwards arrow on top close vertical bar space plus space open vertical bar straight B with rightwards arrow on top close vertical bar greater-than or slanted equal to space open vertical bar open vertical bar straight A with rightwards arrow on top close vertical bar minus open vertical bar straight B with rightwards arrow on top close vertical bar close vertical bar

To show: 

open vertical bar straight A with rightwards arrow on top plus straight B with rightwards arrow on top close vertical bar space equals square root of straight A squared plus straight B squared plus 2 ABcosθ end root


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open vertical bar straight A with rightwards harpoon with barb upwards on top plus straight B with rightwards harpoon with barb upwards on top close vertical bar space less-than or slanted equal to space square root of straight A squared plus straight B squared plus 2 AB end root

space space space space space space space space space space space space space space equals space square root of left parenthesis straight A plus straight B right parenthesis squared end root

space space space space space space space space space space space space space space equals space straight A space plus space straight B

Therefore comma space

open vertical bar straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top close vertical bar less-than or slanted equal to open vertical bar straight A with rightwards harpoon with barb upwards on top space close vertical bar space plus space open vertical bar space straight B with rightwards harpoon with barb upwards on top close vertical bar space

Now comma space open vertical bar straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top close vertical bar space equals space square root of straight A squared plus straight B squared plus 2 ABcosθ end root

Since comma space cos space straight theta space greater-than or slanted equal to space minus space 1

therefore space open vertical bar straight A with rightwards harpoon with barb upwards on top space plus space straight B with rightwards harpoon with barb upwards on top close vertical bar greater-than or slanted equal to space square root of straight A squared plus straight B squared minus 2 AB end root space

space space space space space space space space space space space space space space space space space space space equals square root of left parenthesis straight A minus straight B right parenthesis squared end root

space space space space space space space space space space space space space space space space space space space equals open vertical bar straight A with rightwards harpoon with barb upwards on top space minus space straight B with rightwards harpoon with barb upwards on top close vertical bar

space space space space space space space space space space space space space space space space space space space space equals space open vertical bar open vertical bar straight A with rightwards harpoon with barb upwards on top close vertical bar plus open vertical bar straight B with rightwards harpoon with barb upwards on top close vertical bar close vertical bar

Hence proved. 

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What is a vector quantity?

A physical quantity that requires direction along with magnitude, for its complete specification is called a vector quantity.
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Give three examples of scalar quantities.

Mass, temperature and energy
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What is a scalar quantity?

A physical quantity that requires only magnitude for its complete specification is called a scalar quantity.
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Give three examples of vector quantities.

Force, impulse and momentum.
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What are the basic characteristics that a quantity must possess so that it may be a vector quantity?

A quantity must possess the direction and must follow the vector axioms. Any quantity that follows the vector axioms are classified as vectors. 


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