Analytic Geometry Answers

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

(5.1) 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 >.

(5.2) Find an equation for the plane that contains the line x = −1 + 3t, y = 5 + 3t, z = 2 + t and is (6)

parallel to the line of intersection of the planes x −2(y −1) + 3z = −1 and y = −2x −1 = 0.

(3.1)Find the point of intersection between the lines: < 3, −1, 2 > + t < 1, 1, −1 > and

< −8, 2, 0 > + t < −3, 2, −7 >.

(3.2) 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.

(3.3) Find the point of intersection between 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 >, for t ∈ R.

(4.1) Show that the above line L lies on the plane −2x + 3y − 4z + 1 = 0.

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

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

(3.2) Show that the lines x + 1 = 3t, y = 1, z + 5 = 2t for t ∈ R and x + 2 = s, y − 3 = −5s, (5) z + 4 = −2s for t ∈ R intersect, and find the point of intersection.

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

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