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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember the definition of dot product, such that:

`bar a*bar b = |bar a|*|bar b|*cos(hat(bar a,bar b))`

You also need to remember how to perform the multiplication of two vectors such that:

`bar a = u*bar i + v*bar j`

`bar b = w*bar i + z*bar j`

`bar a*bar b = u*w + v*z`

You need to evaluate the given product, using the definitions stated above, such that:

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

Since` bar u*bar u = |bar u|^2` and `bar v*bar v = |bar v|^2` yields:

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

Hence, evaluating the given product using the definition of dot product, yields the missing coefficients such that: `(bar u - 5bar v)(bar u - bar v) = |bar u|^2 - 6|bar u|*|bar v|*cos alpha + 5|bar v|^2.`

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