Math 11 Summer 2020 Practice Problems for Quiz 2 (in-class quiz); focus sections 12.2-12.5
Disclaimer: This is a sample of representative problems for extra practice and is not comprehensive. Do not expect
the quiz to be completely similar because there are a wide variety of problem types. Be sure that you can do these
problems with understanding and without help.
1)
Given a = 2, 4, â1 and b = 1, â2,3 , find:
(a) 2a â b
2)
(b) a
(c) a b
(d) b ïŽ a
(e) proja b
(f) the angle between a and b
Given two orthogonal vectors u and v , and given the length of u is 4 and the length of v is 5,
compute the following (clearly explain the properties used and show all steps):
2
(a) u ïŽ v
(b) u ï ( u + v )
(c) u + v
1
3)
Use components to prove the following for vectors in V3 . c is a constant. Give reasons for each step.
(a) a ï a = a
4)
2
(b) a ïŽ a = 0
(c)
( ca ) ï b = c ( a ï b )
Let P be a point not on the line L that passes through the points Q and R. Show that the distance d
from the point P to the line L is d =
a ïŽb
a
where a = QR and b = QP .
2
5)
Find parametric equations for the line
(a) through ( 4, â1, 2 ) and (1,1,5 )
(b) through ( â2, 2, 4 ) and perpendicular to the plane 2 x â y + 5 z = 12
6)
(a) Find a linear equation in three variables for the plane determined by the points P ( â5,3, â3) ,
Q ( â3, â1,5 ) and R ( 0, 0, â3) .
(b) Find the area of ïPQR .
(c) Find the angle of vertex P in ïPQR .
3
7)
Find the point in which the line x = 2 â t , y = 1 + 3t , z = 4t intersects the plane 2 x â y + z = 2 .
8)
Find an equation for the plane through (1, 2, â2 ) that contains the line x = 2t , y = 3 â t , z = 1 + 3t
9)
Let the plane P contain the line L : r (t ) = 3t , 2 + 6t , 2t â 3 and the point Q (1, â2,3) . Illustrate the
following with sketches, and find
(a) the distance from Q to L. (use result from #4)
(b) a linear equation of the plane P.
(c) the equation of the line containing Q and normal to the plane P.
(d) the coordinate of the point on line L that is closest to point Q. (this is the projection of Q to L.)
4
10)
Given planes P1 : 2 x + y â 2 z = 5 and P2 : 3x â 6 y â 2 z = â15 . Find:
(a) the angle of intersection of the planes. (b)the vector equation of the line at which the planes intersect.
5
11)
Given two lines L1 : x = 3 + 2t , y = 2 â t , z = â4 + 2t and L2 : x = 5 + 2 s, y = â1, z = s
(a) determine if the lines are parallel, skew, or intersecting. If they intersect, find the point of
intersection.
(b) the equation of the plane containing both lines, if any.
6
12)
Given plane P: 3 x + 6 y + 2 z = 6 . Find:
(a) the distance from Q ( 2, â5,1) to the plane.
(b) the equation of the sphere centered at Q ( 2, â5,1) and tangent to the plane.
(c) the point on P closest to Q ( 2, â5,1) .
7
13)
Determine the magnitude of the resultant force
and its direction measured clockwise from the positive x-axis.
Draw the resultant force on the figure using a parallelogram.
Concept checks.
14) True or false
(a) a ïŽ b = b ïŽ a
(b)
â2, 4, â6 is a normal vector to the plane x â 2 y + 3 z â 5 = 0
(c) You can find a plane that contains two skew lines.
8
(d) Two planes parallel to a line are parallel to each other (in space).
(e) Two planes each perpendicular to a line are parallel to each other (in space).
(f) Two planes either intersect or are parallel.
(g) Two lines either intersect or are parallel (in space).
(h) A plane and a line either intersect or are parallel.
15) Which of the following are meaningful expressions (meaning you can do the computation)?
Answer Yes or No.
(a)
(a ïb ) c
(
(e) a ïŽ b ïŽ c
)
(b)
(a ïb ) ïc
(c) a b + c
(f)
(a ïb )ïŽ c
(g) a ï b ïŽ c
(
(
)
)
(d) a ï b + c
(h) a ïŽ b
16) Fill in the blanks with the best and most complete answer.
(a) a ïŽ a = ______
(
(b) a ï b ïŽ c
(c)
)
represents the ____________ of a _____________________ . (geometrically)
( a ïŽ b ) ï b = ______
(d) a ï a = ______
9
3) [8]
Describe in words and sketch each of the following surface/solid/region, in the 3-dimensional coordinate
system. Use the standard orientation for the axes, with the positive z-axis pointing up.
(a) y2 = 9
(b) xÂČ + y2 +z?
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