- #1

evinda

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Let $W$ be the subspace of $\mathbb{R}^3$ that is orthogonal to the vector $w_1=(-1,-1,1)$ and $p=(x,y,z)$ the projection of the vector $v=(-1,1,2)$ onto $W$. What is $7x-11y+5z$ equal to?I have thought the following:

$\text{proj}_Wv=\frac{\langle v, w_1\rangle}{\langle w_1, w_1\rangle} w_1 \Rightarrow (x,y,z)=\frac{(-1,1,2) \cdot (-1,-1,1)}{(-1,-1,1) \cdot (-1,-1,1)} (-1,-1,1) =\frac{1-1+2}{1+1+1}(-1,-1,1)=\frac{2}{3}(-1,-1,1)=\left( -\frac{2}{3},-\frac{2}{3},\frac{2}{3}\right)$

Then we get that

$$7x-11y+5z=7 \left( -\frac{2}{3}\right)-11\left( -\frac{2}{3}\right)+5\left( \frac{2}{3}\right)=6$$Have I done something wrong? Because the possible answers are $-4,-12,9,15,-14,13$.