Analytic Geometry Answers

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

1.x+y+3z+10=0 and x+2y-z=1,

2.3x-2y+z-6=0 and 4x+2y-4z=0,

3.3x+y+z-1=0 and -x+2y+z+3=0,

4.x-3y+z+1=0 and 3x-4y+z-1=0.

1.let u=<-2,1,-2>,v=<-3,2,-1> and w=<1,3,5>.compute:

1.1.U*W.

1.2. U*(W*V) and (U*W)*V.

2.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.

1.Assume that a vector a of length ||a||=3 units.in addition, a points in a direction that is 135° counter-clockwise from the positive x-axis, and a vector b in the xy plane has a length ||b||=1/3 and points in the positive y-direction.

1.1.find a.b.

1.2.calculate the distance between the point (-1,√3) and the line 2x-2y-5=0.

a.) Consider the point A=(-1,0,1), B=(0, - 2,3) and C = (-4,4,1) to be vertices of a triangle "\\Delta" . Evaluate all side lengths of

"\\Delta"

b.) let "\\Delta" be the triangle with vertices the points P=(3, 1,-1), Q=(2, 0,3) and R=(1, 1,1). Determine whether "\\Delta" is a right triangle. If it is not, explain with reason, why?

c.) let u=<0,1,1>, v=<2,2,0> and w=<-1,1,0> be three vectors in standard form. (i) Determine which two vectors form a right angle triangle? (ii) find "\\theta" =uw, the angel between the given two vectors.

d.) let x<0.find the vector n=<x, Y, z> that is orthogonal to all three vectors u=<1,1,-2>,v=<-1,2,0> and w=<-1,0,1>.

e.)find a unit vector that is orthogonal to both u =<0,-1,-1> and v=<1,0,-1>

1.consider the point a=(-1,0,1), b=(0,-2,3),and c=(-4,4,1) to be vertices of a triangle∆.evaluate all side lengths of ∆.

2.let ∆ be the triangle with vertices the points p=(3,1,-1),q=(2,0,3)and r=(1,1,1).determine whether ∆ is a right angle triangle.if it is not ,explain with reason,why?

Determine proj_au the orthogonal projection of u and a and deduce ||proj_au|| for

a.) u=<-1 3>, a=<-1,-3>;

b.)u=<-2,1,-3>, a=<-2,1,2>.

Assume that A vector a of length ||a|| =3 units. In addition, a points in a direction that is 135° counterclockwise from the positive x-axis, and a vector b in the xy-plane has a length ||b|| =1/3 and points in the positive y-direction.

a.) Find a.b

b.) calculate the distance between the point(-1,3) and the line 2x - 2y-5=0

(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 = 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,

(d) x − 3y + z + 1 = 0 and 3x − 4y + z − 1 = 0.

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.

Assume that a vector ~a of length ||~a|| = 3 units. In addition, ~a points in a direction that is 135◦

counter-

clockwise from the positive x-axis, and a vector ~b in th xy-plane has a length ||~b|| =

1

3

and points in the

positive y-direction.

(5.1) Find ~a · ~b.

(5.2) Calculate the distance between the point (−1, √

3) and the line 2x − 2y − 5 = 0.