91Ó°ÊÓ

To find a point on the line of intersection, we need to solve the system of two equations formed by the given planes Q and R: $$ Q: x-y-2z=1 \\ R: x+y+z=-1 $$ To solve this system, we can add the two equations to eliminate the y variable: $$ Q+R: x-y-2z+x+y+z = 1-1 \\ 2x-z = 0 $$ Now we can assign an arbitrary value for one of the variables left (let’s choose \(z=t\)) and find the corresponding values for the other variables: $$ x = \frac{1}{2}z \\ x = \frac{1}{2}t $$ Now we have the parametric representation of the variables x and z. This implies that for any value of \(t\), we can find the values of x and z that lie on the line of intersection. To find the corresponding y value: $$ y = z - x - 1 \\ y = t - \frac{1}{2}t - 1 \\ y = \frac{1}{2}t - 1 $$ So the parametric representation of the point on the line of intersection is: $$ P(t) = (\frac{1}{2}t, \frac{1}{2}t - 1, t) $$

To find the direction vector of the line of intersection, we need to find the cross product of the normal vectors of the two planes. The normal vectors of planes Q and R are given by the coefficients of the variables in the equations: $$ \vec{N_Q} = (1, -1, -2) \\ \vec{N_R} = (1, 1, 1) $$ Now we compute the cross product \(\vec{N_Q} \times \vec{N_R}\): $$ \vec{d} = \vec{N_Q} \times \vec{N_R} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & -2 \\ 1 & 1 & 1 \end{vmatrix} \\ \vec{d} = (3\hat{i} - 3\hat{j} + 2\hat{k}) $$ So the direction vector of the line of intersection is \(\vec{d} = (3, -3, 2)\).

Now, we have the parametric representation of a point P(t) on the line of intersection and the direction vector \(\vec{d} = (3, -3, 2)\). The parametric equation of the line can be written as: $$ \vec{r}(t) = P(t) + t \vec{d} \\ \vec{r}(t) = (\frac{1}{2}t, \frac{1}{2}t - 1, t) + t(3, -3, 2) \\ \vec{r}(t) = (\frac{1}{2}t + 3t, \frac{1}{2}t - 1 - 3t, t + 2t) \\ \vec{r}(t) = \left( \frac{7}{2}t, \frac{-5}{2}t - 1, 3t \right) $$ So the parametric equation of the line of intersection is: $$ \vec{r}(t) = \left( \frac{7}{2}t, \frac{-5}{2}t - 1, 3t \right) $$