Question

problem 4 (fill in the blank) : write the vector equation and the parametric equations for the plane containing the points, $p = (1, 3, -1)$, $q = (-2, 1, 1)$, $o = (2, -3, 2)$. hint : use the vectors $overrightarrow{pq}$ and $overrightarrow{po}$. for full credit, please show the relevant calculations. vector equation : <your answer here> parametric equations : <your answer here>

Explanation:

Step1: Calculate $\overrightarrow{PQ}$

$\overrightarrow{PQ} = Q - P = (-2-1, 1-3, 1-(-1)) = (-3, -2, 2)$

Step2: Calculate $\overrightarrow{PO}$

$\overrightarrow{PO} = O - P = (2-1, -3-3, 2-(-1)) = (1, -6, 3)$

Step3: Write vector equation

Let $\mathbf{r} = (x,y,z)$ be a point on the plane, $\mathbf{r_0} = P = (1,3,-1)$. The vector equation is:
$$\mathbf{r} = \mathbf{r_0} + s\overrightarrow{PQ} + t\overrightarrow{PO}$$
Substitute values:
$$(x,y,z) = (1,3,-1) + s(-3,-2,2) + t(1,-6,3)$$

Step4: Derive parametric equations

Break the vector equation into x, y, z components:
$x = 1 - 3s + t$
$y = 3 - 2s - 6t$
$z = -1 + 2s + 3t$
where $s,t \in \mathbb{R}$

Answer:

vector equation: $\boldsymbol{(x,y,z) = (1,3,-1) + s(-3,-2,2) + t(1,-6,3)}$ (where $s,t$ are real numbers)
parametric equations:
$x = 1 - 3s + t$
$y = 3 - 2s - 6t$
$z = -1 + 2s + 3t$ (where $s,t$ are real numbers)