Analytic Geometry Answers

(2.1) 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 >.

(2.2) Find parametric equations of the line that passes through the points A = (1, 2, −3) and B =

(7, 2, −4).

(2.3) Find parametric equations for the line of intersection of the planes −5x + y − 2z = 3 and

2x − 3y + 5z = −7.

(1.1) Determine whether the given line and the given plane are parallel:

(a) x = 1 + t, y = −1 − t, z = −2t and x + 2y + 3z − 9 = 0,

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

Which one of the following describes the points that satisfy the equation

"(x;y;z) = (-1;2;0) + \\lambda" "(1,3,-2)"

1. A plane through the point "(-1,2,0)" perpendicular to the line "r = \\lambda(1,3,-2)"
2. A plane through the point "(-1,2,0)" parallel to the one "r = \\lambda(1,3,-2)"
3. A sphere centered at "(-1,-3,2)" with a radius 5
4. A line through the point "(-1,2,0)" with direction vector "d = \\lambda(1,3,-2)"

1.     Find a point-normal form of the equation of the plane passing through P = (1, 2,-3) and having n =< 2,-1, 2 > as a normal.

2.     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

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

b)     x - 3y + z + 1 = 0 and 3x - 4y + z - 1 = 0

Question 6

Let u =< −2,1,−1,v =< −3,2,−1 >and w =< 1,3,5 >. Compute:

(6.1) u ×w,

(6.2)u ×(w ×v)and(u × w)×v.

Question 7

(7.1) (Find a point-normal form of the equation of the plane passing

through P = (1,2,−3) and having n =< 2,−1,2 > as a normal.

(7.2) Determine in each case whether the given planes are parallel or

perpendicular: (a) x +y +3z +10=0andx +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,

(d)x −3y +z+1=0 and 3x −4y +z−1=0.

Find an expression for 12

jj~u + ~vjj2 + 12

jj~u 􀀀 ~vjj2 in terms of jj~ujj2 + jj~vjj2.

(3.2) Find an expression for jj~u + ~vjj2 􀀀 jj~u 􀀀 ~vjj2 in terms of ~u
~v (3)

(3.3) Use the result of (3.2) to deduce an expression for jj~u + ~vjj2 whenever ~u and ~v are orthogonal (1)

to each other.

Determine projau the orthogonal projection of u and a and deduce ||projau|| for (2.1) u =<−1,3 >, a=<−1,−3>;

(2.2) u =<−2,1,−3 >, a =<−2,1,2 >.

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

(1.2) .u =<1,−2,4 >, v =<5,3,7 >.