Calculating the inner product of vectors u and v.

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I have vectors:

$u$ = $(4, -2, 2, 5)$

$v$ = $(4, 1, 4, 2)$

The dot product is defined as:

$(u, v)$ = $u_1$$v_1$ + $u_2$$v_2$ + $u_3$$v_3$ + $6$$u_4$$v_4$

Using the above information, I've calculated the above dot product to be 82.

However, I'm asked to calculate the inner dot product:

$(u - v, u + 2v)$

How do I go about doing this?

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2 Answers

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$u-v=(0,-3,-2,3)$. And $u+2v=(12,0,10,9)$. Now dot them. Get $142$.

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\begin{align} \langle u-v,u+2v\rangle & =\langle (4,-2,2,5)-(4,1,4,2),(4,-2,2,5)+(8,2,8,4)\rangle \\ & = \langle (0,-3,-2,3),(12,0,10,9)\rangle \\ & = 0+0-20+6*27=-20+162=142. \end{align}

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