Show that the scalar product of two vectors is equal to the sum of the products of their corresponding rectangular components.

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Solution

Let there be two vector $\to a$ and $\to b$ subtending an angle $θ$ , and $θ_{2}$ with horizontal respectively.

Thus by scalar product formula,

$\to a . \to b = a b cos (θ_{2} - θ_{1})$ ...(1)

Now by protection their rectangular components

$\to a = | \to a | cos θ_{1} + | \to a | sin θ_{1}$

$\to b = | \to b | cos θ_{1} + | \to b | sin θ_{2}$

Now,

$\to a . \to b = (a cos θ_{1} + a sin θ_{1}) (b cos θ_{2} + b sin θ_{2})$

$\to a . \to b = a b cos θ_{1} . cos θ_{2} + a b sin θ_{1} . sin θ_{2}$

$\to a . \to b = a b (cos θ_{1} . cos θ_{2} + sin θ_{1} . sin θ_{2})$

$\to a . \to b = a b cos (θ_{2} - θ_{1})$ proved

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