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Determine whether the points $$P$$ and $$Q$$ lie on the given surface.
$$\mathbf{r}(u, v)=\langle u+v, u-2 v, 3+u-v\rangle \quad P(4,-5,1), Q(0,4,6)$$

$$r\left(u, v\right)=\left\langle u+v, u-2 v, 3+u-v\right\rangle$$

$$P(4,-5,1)$$
$$Q(0,4,6)$$

(1) $$P(4,-5,1)$$

$$u+v=4 \quad, \quad u-2 v=-5 \quad, \quad 3+u-v=1$$

$$3 v=9 \quad v=3, u=1$$

check: $$3+u-v=3+1-3=1$$ #

$$p$$ on the surtice

(2) $$Q(0,4,6)$$

$$u+v=0 \quad , \quad u - 2 v=4 \quad , \quad 3+u-v=6$$

$$u=\frac{4}{3} \quad v=\frac{-4}{3} $$

$$3+\frac{4}{3}-\left(-\frac{4}{3}\right) \neq 6$$

$$Q$$ doesn't lie on surface

Identify the surface with the given vector equation.
$$\mathbf{r}(u, v)=(u+v) \mathbf{i}+(3-v) \mathbf{j}+(1+4 u+5 v) \mathbf{k}$$

$$r(u, v)=(u+v) i+(3-v) j+(1+4 u+5 v) k$$

$$a_{0}=\left(0 , 3 , 1\right)$$

$$a=(1,0,4)$$

$$b=(1,-1,5)$$

$$a \times b=\left|\begin{array}{ccc}{i} & {j} & {k} \\ {1} & {0} & {4} \\ {1} & {-1} & {5}\end{array}\right|=(0+4) i+(5-4) j+(-1) k$$

$$=4 i-j-k$$

equation of plan

$$4[x-0]-1[y-3]-1[z-1]=0$$

$$4 x-y-z=-4$$

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