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PortraitPro Studio Crack Mac Activation Free Download.Q:
Confused with how the proof of the sandwich theorem goes
Consider the plane $x + y – z = 2$ and a point $\mathbf v$. We know that if the line through $\mathbf v$ and $\mathbf 0$ contains the point $(\lambda,\lambda,\lambda)$ for some $\lambda \in \mathbb R$, then $\mathbf v$ lies on the plane.
We also know that
$\lambda(\mathbf v + \mathbf 0) + (1-\lambda)(\mathbf v + \mathbf 0) – \mathbf v = \lambda\mathbf v + (1-\lambda) \mathbf 0$
How does this equal $\mathbf 0$? Clearly $\mathbf 0 + \mathbf 0$ is even no longer $0$, and since $\mathbf v$ is the third entry, but what does $\mathbf 0$ equal? Or, is this some way of saying that the line is perpendicular to the plane? I’m not totally sure what this is being used for, but surely it’s not one of the usual suspects.
A:
The claim is that $(\lambda\mathbf v + (1-\lambda)0)$ lies in the plane, and thus in particular, $\lambda\mathbf v+ (1-\lambda)0$ is in the plane.
The big point is that this relation only holds for scalars $\lambda$ that sum to $1$ (so that your line is perpendicular to the plane).
To see why this works, you first realize that if the point $(x_1,x_2,x_3)$ lies in the plane, then $x_2 = x_3$, and therefore $\lambda(x_1+0)+ (1-\lambda)(x_1+0) – x_1 = 0$ for all $\lambda$. We now plug in the point $(x_1,x_2,x_3)$ and see that this holds true. That is, we know $x_1 + x_2 – x_3 = 0$.
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