Analytic Geometry Answers

Find the vector form of the equation of the plane that passes through the point P₀ = (1, - 2,3) and has normal vector n =<3,1,-1>.

Find an equation for the plane that contains the line x = - 1 +3t, y=5 +3t, z =2 +t and is parallel to the line of intersection of the planes x-2(y - 1) +3z =-1 and y =-2x - 1 =0

a.) Find the point of intersection between the lines :<3, - 1,2> +<1, 1, - 1> and <-8, 2, 0> +t <-3,2-7>.

b.) show that the lines x +1 =3t, y=1, z +5 = 2t for t€ R and x +2 =s, y-3 = - 5s, z +4=-2s for t € R intersect, and find the point of intersection.

c.) Find the point of intersection between the planes : - 5x + y - 2z =3 and 2x - 3y +5z =-7.

D.)let L be the line given by <3, - 1,2> +t<1,1-1>, for t € R.

1.) show that the above line L lies on the plane - 2x + 3y - 4z +1 =0

2.)Find an equation for the plane through the point P =(3, - 2,4)that is perpendicular to the line <-8, 2, 0> +t<-3,2,-7>

1.1 find parametric equations of the line that passes through the point P =(2, 0,-1) and is parallel to the vector n =<2, 1, 3>

1.2 find paramedic equations of the line that passes through the points A= (1, 2, - 3) and B =(7, 2, - 4).

1.3 find paramedic equations for the line of intersection of the planes - 5x + y - 2z =3 and 2x - 3y + 5z =-7

Find the vector form of the equation of the plane that passes through the point P0 = (1, −2, 3) (5)

and has normal vector ~n =< 3, 1, −1 >.

Determine whether ~u and ~v are orthogonal vectors, make an acute or obtuse angle: (1.1) ~u =< 1, 3, −2 >, ~v =< −5, 3, 2 >. (2) (1.2) ~u =< 1, −2, 4 >, ~v =< 5, 3, 7 >.

Determine in each case whether the given planes are parallel or perpendicular

a) x + y + 3z 10 = 0 and x + 2y - z = 1

b) 3x - 2y + z - 6 = 0 and 4x + 2y - 4z = 0

c) 3x + y + z - 1 = 0 and -x + 2y + z + 3 = 0

Let vector "a = (1, sqrt(2) ,1)" and vector "b = (2,0,2)"

1. Find the angle "\\theta" between the 2 vectors.
2. Find the projection of a in the direction of b

Which of the following planes are parallel to the line with equation "r = (3,0,1) + \\lambda(3,2,-1)"

1. "2x - 3y = 6"
2. "x - y+z=6"
3. "x - 27 + z = 6"
4. "3 y + z = 6"

Let the plane P pass through the 3 points R = (1, 1, 6), S= (2,5,4) and T = (1,2,3).

Find the values of a, b, c and d in the equation "ax + by + cz = d" of plane P.