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The sum of two vectors
u
and
v
yields
What is the dot product of two perpendicular vectors?
Which vector operation is most helpful when trying to find the area of a
parallelogram?
Which vector operation is most helpful when trying to find the projection of one
vector onto another?
The cross product satisfies all of the following criteria EXCEPT
What happens when a vector is multiplied by a scalar?
What is the sum of the vectors that point to the vertices of a cube centered at
the origin (in 3dim space)?
Which of the following strategies is useful in answering the previous question?
Which of the following could not be the sum of two 3dimensional vectors
in the
x

y
plane?
Which of the following could not be the cross product of two 3
dimensional vectors in the
x

y
plane?
Which of the following is not a similarity between the dot product and the cross
product?
What is
(1, 2, 5) + 3(1, 0,  1)
?
What is
(1, 2, 5)·(1, 0,  1)
?
What is
(1, 0, 1)×(0, 14, 0)
?
What is
(1, 0, 1)·(0, 14, 0)
?
What is the
x
component of the cross product
(13, 23, 33)×( 3, 0, 0)
?
What is
(1, 0)×(0, 1)
?
The displacement of a man walking through the desert can be represented
by a vector: the magnitude of the vector corresponds to the distance he
has walked, while the direction of the vector corresponds to the
direction in which he has walked. (This is perhaps the most intuitive example
of vectors being applied to real problems). The next 5 questions will deal with
such displacement vectors.
If your dog's displacement vector (in a place where your house sits at the
origin) is given by
(3, 5)
, how far is she from home?
If a man walks 3 miles east, then 4 miles north, what is his final displacement
vector (in miles)?
If a rabbit makes 3 successive displacements, given by the vectors
u
,
v
, and
w
, what is its total displacement vector from the starting point?
Ernesto Mojito is trying to find his way home late at night. He should have
walked in the direction of the unit vector
u
, but ended up at a
displacement
v
from his original starting point. How far has he gone in the
right direction?
Let El Palacio Real be at the origin of Madrid. Assume El Museo del
Prado lies (in kilometers) at
(4, 2)
, and Metropolis lies at
(3, 0)
. How far must a tourist walk to go from Metropolis to Prado? (Hint:
first find the displacement vector using vector subtraction, then compute its
magnitude.)
The velocity of a moving car can be represented by a vector: the
magnitude of the vector corresponds to the speed of the car, the
direction of the vector corresponds to the direction in which the car is
moving. If the car speeds up or slows down, the magnitude of its corresponding
velocity vector changes (gets longer or shorter). If the car turns, the
direction of its velocity vector is also altered (and will rotate to point in
the new direction in which the car is heading). This idea will be useful in
answering the following 5 questions.
If a car has initial velocity vector
v
and then doubles in speed, what is the
car's new velocity vector?
If a car with initial velocity vector
(1, 0)
turns 90 degress to the right,
without changing its speed, what is its new velocity vector?
What is the speed of a car with velocity
(3, 4)
?
A car with initial velocity
(3, 4)
makes a 37 degree turn to the left, drives
30 minutes in this direction and then turns 85 degrees to the right. After
another 20 minutes of travel, the car turns again, and it is now unclear in
which direction the car is traveling. However, the car is still traveling at
its original speed. Which of the following is a candidate for the present
velocity of the car?
A car with initial velocity
(1, 0, 0)
gets driven off a cliff. Which of the
following is a candidate for the car's final velocity as it hits the ocean?
(Assume the positive
z
direction points upward to the sky).
A line passing through the origin in 3dimensional space can be characterized by
a vector
v
: all points on the line can be written in the form
t
v
, where
t
is a real number (
t = 0
yields the origin itself), and
the full set of points (ranging over all values of t) is the whole line. For a
line that does not pass through the origin, all the points can be written of the
form
u + t
v
, where
u
is a particular vector which can be
chosen at will from the points which lie on the line. This idea is central to
the following 5 questions.
Which of the following is a set of two distinct parallel lines?
Which of the following is a point on the line given by
(1, 2, 1) + t(3, 0,  1)
?
If one has two lines, given by
u
_{1} + t
v
_{1}
and
u
_{2} + t
v
_{2}
, which condition ensures that they will be parallel?
If one has two lines, given by
u
_{1} + t
v
_{1}
and
u
_{2} + t
v
_{2}
, which condition ensures that they will be
perpendicular?
In the plane, which line is NOT the same as
(1, 0) + t(2, 3)
?