Download Question 1: Given the vectors = (3,2,1) , = (0,1,–1) , and = (–1, 1,0

Question 1:
Given the vectors v = (3,2,1) , u = (0,1,–1) , and w = (–1, 1,0) , compute the following :
a) The unit vector parallel to w
b) The angle between v and w
c) The vector projection of u on w
d) The equation of the plane parallel to v and w through the origin
e) The equation of the line parallel to v through (1,0,0). Does this line pass through the
point (–5,–4,–2)?
Question 2:
Given the plane 2x – 2y + z = 4 and the point P1 (0,–1, 2) on it. Let v = (1,0,1) be a vector in
component form.
a) Let the initial point of v be P1 . Give the coordinates of its terminal point Q .
b) Give the component form of a vector perpendicular to the plane.

c) Let P2 be a point on the plane such that the vector P2 Q is perpendicular to the plane. What

is P2 Q ?
d) What are the coordinates of P2 ? (Hint: The point P2 is the intersection of the line through
P2 and Q with the plane)
Question 3:
A sphere with radius 2 is centered at the origin. Find the parametric equation of the
intersection of the tangent planes to the sphere at ( 1, –1, 0) and ( -1, 0, 1).
Question 4:
Find all the local maxima, minima, and saddle points of the function
f(x,y) = 2 x 3  2 y 3  9 x 2  3 y 2  12 y .
Question 1:
A sphere with radius 2 is centered at the origin. Find the parametric equation of the
intersection of the tangent planes to the sphere at ( 0, 0, 2 ) and ( -1, 0, 1).
Question 1:
Find all critical points of the function f ( x, y)  ( x 2  y 2 )( x  1) and classify them.