Use the properties of dot product to rewrite ((vector u)-(5(vector v))) dot product ((vector u)-1(vector v))     as ____?___(length of vector u)^2 + ____?____( (vector u) dot product (vector v))+ ___?___(length of vector v)^2.

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You need to perform the multiplication of the vectors `(bar u - 5 bar v)*(bar u - bar v)` such that:

`(bar u - 5 bar v)*(bar u - bar v) = bar u*bar u - bar u*bar v - 5 bar v*bar u + 5 bar v*bar v`

`(bar...

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You need to perform the multiplication of the vectors `(bar u - 5 bar v)*(bar u - bar v)` such that:

`(bar u - 5 bar v)*(bar u - bar v) = bar u*bar u - bar u*bar v - 5 bar v*bar u + 5 bar v*bar v`

`(bar u - 5 bar v)*(bar u - bar v) = bar u*bar u - 6 bar u*bar v + 5 bar v*bar v`

Using the definition of dot product yields:

`(bar u - 5 bar v)*(bar u - bar v) = |bar u|^2*cos (hat(bar u, bar u)) - 6|bar u|*|bar v|*cos (hat(bar u, bar v)) + 5 |bar v|^2cos (hat(bar v, bar v))`

Since `cos (hat(bar u, bar u)) = cos (hat(bar v, bar v)) = cos 0^o = 1` yields:

`(bar u - 5 bar v)*(bar u - bar v) = |bar u|^2 - 6|bar u|*|bar v|*cos (hat(bar u, bar v)) + 5 |bar v|^2`

Hence, performing the multiplication of the vectors yields `(bar u - 5 bar v)*(bar u - bar v) = |bar u|^2 - 6|bar u|*|bar v|*cos (hat(bar u, bar v)) + 5 |bar v|^2.`

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