Analytic Geometry Answers

a) Find a non-zero unit vector u with terminal point Q(-7,1,4) such that u has the opposite direction as v=(3,2,1).

b) Find a non-zero unit vector u with terminal point Q(-7,1,4) such that u has the same direction as v=(1,2,3).

Find the vector equation of the plane determined by the points (1, 0, −1),
(0, 1, 1) and (−1, 1, 0). Alsofind the point of intersection of the line

Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0). Also nd the point of intersection of the line

Show that the closed sphere with centre )7,3,2( and radius 10 in R³ is contained in the
open cube P = {(x, y,z :) x − 2 <11, y − 3 <11, z − 7 <11}.

Does there exist a plane targent to x²-2y²-2z²=8 and which passes through
2x+3y+2z = 8, x−y+2z = 5? Justify your answer.

Find the equation of the plane passing through the line of intersection of the planes
2x+3y+z = 4 and x+y+z = 2, and which is perpendicular to the plane
2x+3y−z = 3.

Does there pass a plane through the lines x+4/3 =
y/2 =
z−1/3
and x/2 =
y−1/1 =
z+1/1
?
Justify.

Find the equation of the right circular cone whose vertex is (1,0,1), the axis is
x−1 = y−2 = z−3, and the semi-vertical angle is 30◦
. Also, find the section of
the cone by the coordinate planes.

Reduce the following equations to standard form, and then identify which conicoids
they represent. Further, give a rough sketch of each.
i) x^2 +y^2 +2x−y−z+3 = 0
ii) 3y^2 +3z^2 +4x+3y+z = 9

Show that x = y = z+1 is a secant line of the sphere x
2 +y
2 +z
2 −x−y+z−1 = 0.
Also find the intercept made by the sphere on the line.