PROJECT $1$ DUE: FRIDAY February $2.$ Make sure to annotate your work in text mode. A series of pictures and calculations without discussion will not receive full credit.

Two planes are parallel if their normal vectors are parallel. Consider the planes:

$x+2y-3z=4$ and $2x+4y-6z=3$

a$)$ What are the normal vectors $(n1,n2)?$

b$)$ Are they parallel? Why or why not?

c$)$ Use the plot$3$d function to plot both planes on the same graph. Rotate the picture to make sure you are convinced of your answer.

$2.$ If two planes intersect, then the angle between them is the same as the angle between their normal vectors and can be found in the usual way using the dot product definition for $cos()=\frac{u*v}{|u||v|}.$ Remember to use with$($LinearAlgebra$)$: first.

a$)$ Find the angle between the planes $x+y+z=1$ and $x-2y+3z=1.$

b$)$ Plot the graphs.

c$)$ Find the parametric and vector equations for the line of intersection $L$ : First find a common point that lies on both planes by letting $z=0,$ for example, in both equations and solving for $x$ and $y.$ Now since $L$ lies in both planes, it is perpendicular to both normal vectors. So the direction vector for $L$ is $v=n1x2.$ Now you have a point and a direction vector. Find the equations.

d$)$ Add $L$ to the graph in part $($b$).$

$3.$ Given the parametric equations for $2$ lines, determine whether the lines are parallel, intersect $($where$?)$ or are skew $($why$?)$

a$)$ L$1$: $x=2t-1,y=-4t+10,z=t$ and L$2$: $x=-5s-12,y=3s+11,z=-2s-4$

b$)$ L$1$: $x=-4t+4,y=3t+2,z=-1+8t$ and $L2$:$x=2t+2,y=6t+2,z=t-3$