Equation of a Plane

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to the straight lines  $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-3}{1}\$  and  $\frac{x+3}{2}=\frac{y-3}{-5}=\frac{z+1}{-3}$  

•  $\overrightarrow{r\ \ }\cdot\left(2i+j+4k\right)=20;\ 2x+y+4z-20=0$ 

•  $\overrightarrow{r\ \ }\cdot\left(i-2j+4k\right)=20;\ x-2y+4z-20=0$ 

•  $\overrightarrow{r\ \ }\cdot\left(i-2j+4k\right)=10;\ x-2y+4z-10=0$ 

•  $\overrightarrow{r\ \ }\cdot\left(3i+2j+3k\right)=20;\ \ 3x+2y+3z-20=0$ 

Find the equation of the plane that goes through the three points A(0, 3, 4), B(1, 2, 0), and C(−1, 6, 4).

• $x=1+3t,\ \ \ \ \ \ y=0−2t,\ \ \ \ \ \ z=−2+5t.$

• $x=3+t,\ \ y=-2,\ \ \ z=5-2t.$

• $x=1+3t,y=1−2t,z=−2+4t.$

• $x=3t,y=0−2t,z=−2+5t.$

Find the equation of the plane that is orthogonal to the line  $x=4+t,\ y=1−2t,\ z=8t$   and goes through the point  $P(3,2,1).$  

•  $x+2y+8z−7=0.$  

•  $3x−4y+8z−7=0.$  

•  $x−2y+8z−7=0.$  

•  $3x−4y+8z+7=0.$  

Find the equation of the plane that is parallel to the plane  $5x−3y+2z=6$    and goes through the point  $P(4,−1,2).$  

•  $4(x+5)+1(y-3)-2(z+2)=0$  

•  $4(x−5)−1(y+3)+2(z−2)=0$  

•  $5(x+4)+3(y-1)-2(z+2)=0$  

•  $5(x−4)−3(y+1)+2(z−2)=0$  

Find the intersection of the plane  $3y+z=0$   and the sphere $x^2+y^2+z^2=4$ .

• All point that satisfy  $x^2+10y^2=4$  

• All point that satisfy  $x^2+y^2+z^2=4$  

• All point that satisfy  $3y+z=0$  

•  $(0,1,-3)$  

Find the equation of the plane that contains the intersecting lines  $x=4+t_1,\ y=2t_1,\ z=1−3t_1$   and  $x=4−3t_2,\ y=3t_2,\ z=1+2t_2.$  

•  $-13x-7y-9z+61=0$  

•  $13x+7y+9z+61=0$  

•  $13x+7y-9z−61=0$  

•  $13x-7y+9z−61=0$  

 Find the equation of the plane that is orthogonal to the plane  $3x+2y−z=4$   and goes through the points   $P(1,2,4)$  and  $Q(−1,3,2).$  

•  $3(x+1)−8(y+2)−7(z+4)=0$  

•  $3(x−1)+8(y−2)−7(z−4)=0$  

•  $3(x−1)−8(y−2)−7(z+4)=0$  

•  $3(x−1)−8(y−2)−7(z−4)=0$