Equation of plane containing straight line?

Question

3


Find the equation of plane containing straight line

"[[[x],[y],[z]]]=[[[1],[0],[-1]]]+t*[[[2],[1],[0]]] and the point P(0,1,2). "

Answer 1

If I understand your notation correctly, you are asking for the equation of a plane containing the line L, where

L = <0,1,2> + t<1,0,-1>

This is not enough to define a unique plane. A plane containing the line could be rotated around it. You need another vector or at least another point to define a unique plane.

Answer 2

Take any 2 points on the line, at t=0 and at t=1 for example; then A(t=0) = [1,0,-1] B(t=1) = [3,1,-1]; take vector AP = (0-1,1-0,2+1) = (-1,1,3); now take vector
AB = (3-1,1-0,-1+1) = (2,1,0); find cross product N=[AP*AB],
N being normal to the plane to be found; or
````| i j k |
N=|-1 1 3 | =-3i + 6j – 3k
````| 2 1 0 |
any vector parallel to N will do, so take N=(1,-2,1);
equation of plane: x-2y+z=D, plug in point A, B or P and find D; D=0;
your plane is: x-2y+z=0 passing through point [0,0,0]

Answer 3

One way is to compute the slopes directly using m=(y1-y2)/(x1-x2). With the first set of points we have (2-(-4))/(-4)=-6/4=-3/2 and the second set gives us m=(3-(-3))/(4-(-5))=6/9=2/3. these are negative reciprocals so they're perpendicular. Another way is to fit these pairs into the point-slope formula for straight lines - y=mx+b. Where m is the slope and b is the y-intercept. Then use the fact that parallel lines have identical slopes while perpendicular lines have slopes which are negative reciprocals of each other. With (0,-4) we have -4=0m+b so b is -4. With (-4,2) we have 2=m(-4)-4 so m=-6/4=-3/2. Now we have y=(-3/2)x-4. Checking: 2=(-6/4)(-4)-4 = 6-4=2 OK and -4=0-4 OK For the other pair we have 1) 3=4m+b and 2) -3=-5m+b and subtracting the equations we get 6=9m m=6/9=2/3 Notice we don't have to go any further since we know the slope. Now observe that the -3/2 slope of the first equation is the negative reciprocal 2/3 so the lines are perpendicular. But I like to check my work so let's continue with the 2nd equation. Using (4,3) we have 3=(2/3)4+b so b=1/3. Now use (-5,-3) to check. -3=(2/3)(-5)+1/3=-10/3+1/3=-3 OK

Answer 4

complex stuff. look into on google. this can assist!