Analytic Geometry Answers

With given vectors u=<3,-1,-2> v=<-1,0,2> w=<-6,1,4> compute the expressions below
2.1 v+3w
2.2 u-2v
2.3 - (v+3u)

find the coordinates of the centre of a circle and its radius if the equation is 1) x²+y²-10x-12y+6=0

2) -2x²-2y²-18y+9=0

Consider the vectors u=<-2,2, - 3>, v=<-1,3, - 4>, w=2, - 6,2> and the points A (2,6,-1) and B(-3, - 5,7). Evaluate

a.) The distance between the two points

b.) ||2u - 3v + 1/2w||

c.) The unit vector in the direction of w.

d.) Suppose u; v and w are vectors in 3D,where u=(u_{1, }u₂, u₃) ; v = (v₁, v₂, v₃) and w=(w₁, w₂, w₃).

Express (u x v). W as a determinant.

We assume given a plane u passing by the tip of the vectors u=<-1,1,2, v=<2, - 1,0> and w=<1, 1, 3.

a.) Find the dot products u.v and w.v

b.) Determine whether or not there is a vector n that is perpendicular to u. If yes, then find the vector n. Otherwise explain why such a vector does not exist?

Knowing the fact that the cross product of two vectors uxv is orthogonal to both vectors u and v, find a case where this is not applicable.

Given the points (-4,8) and (6, -12):

(i) Determine the midpoint of the line segment connecting the points.

(ii) Determine the distance separating the two points.

Prove that the dot product between two vectors is commutative and not associative

let U=(u1,u2) AND V=(v1,v2) belong to R2 verify that <u,v> =u1v1-2u1v2-2u2v1+5u2v2 is an inner product space on r2

reduce x1-2y+3z-4x2x3+6x1x3 to canonical forms