Analytic Geometry Answers

Find the vector form of the equation of the plane passing through the point P (1-23) and has normal vector n <3,1, - 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

Let L be the line given by (3,-1,2) + t (1,1,-1).Show that the above line L lies on the plane -2x + 3y -4z + 1 = 0

1.   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 >.

2.   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.

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

4.   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.

5.   Find the point of intersection between the planes: −5x + y −2z = 3 and 2x −3y + 5z = −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

Determine whether the given line and the given plane are parallel :

a.) x = 1 + t, y=-1, z=-2t and x = 2y +3z - 9 =0,

b.) <0, 1, 2> +t <3,2,-1> and 4x - 2z +1 = 0

(2.1) Find the components of a unit vector satisfying ~v· < 3, −1 >= 0.

(2.2) Show that there are infinitely many vectors in R (4) 3 with Euclidean norm 1 whose Euclidean inner product with < −1, 3, −5 > is zero.

(2.3) Determine all values of k so that ~u =< −3, 2k, −k > is orthogonal to ~v =< 2, (3) 5 2 , −k >

(1.1) Let U and V be the planes given by: U : λx + 5y − 2λz − 3 = 0,

V : −λx + y + 2z + 1 = 0.

Determine for which value(s) of λ the planes U and V are:

(a) orthogonal,

(b) Parallel.

(1.2) Find an equation for the plane that passes through the origin (0, 0, 0) and is parallel to the (3) plane −x + 3y − 2z = 6.

(1.3) Find the distance between the point (−1, −2, 0) and the plane 3x − y + 4z = −2.

Suppose u; v € V and ||u|| = ||v|| = 1 with < u; v > = 1: Prove that u = v