Review of Section 8.5, 8.6, and Chapter 10
Math 1113 (Ritter)
(1) A golf ball is driven into the air with a speed of 75 miles per hour at an angle of 50◦ with respect
to the horizontal. Express this velocity in vector form.
(2) Evaluate the expression or perform the indicated operations with the given vectors.
u = 3i − 2j,
v =< 4, 4 >,
z = 2i + 5j,
w=
√
2 < cos
3π
3π
, sin
>
4
4
(a) 2u + 3w
(b) −5j − 2v
(c) kwk
(d) z · v
(e) projv u
(f) Find any vector that is orthogonal to z
(3) A tow truck pulls a car behind it exerting a force of 1000 pounds on the attachment point. The
attachment point is 45◦ to the horizontal. Find the work done if the truck pulls the car 1 mile. (1 mile
= 5280 ft. Round to the nearest ft lb.)
(4) A truck with a gross weight of 30, 000 pounds is parked on a 7◦ . Find the pounds of force required
to prevent the truck from rolling down the incline. Round to the nearest pound.
(6) Express the vector u =< 4, −2 > as the sum of two vectors, one of which is parallel to v and one
which is perpedicular to v where v =< 1, 3 >.
(7) Determine if the given vectors are parallel, perpendicular, or neither.
(a) ~u =< 0, −3 >, ~v =< 47, 0 >
(b) ~x =< 3, −2 >, ~v = −1, 23
(c) ~u =< 2, −7 >, ~v =< 7, −2 >
(8) Use the given matrices to evaluate the desired expression. If the expression is undefined, state this.


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−3
0
10
1
1
4
1
0
0
4


A=
B= 3
D=
3
1  C=
7
−2
0
−2
1
−2
6
0
0
6
(a) BC
(b) D + 3A
(c) det(3A)
(d) det(C)
(e) DC
(f) CB
(g) det(B)
(9) Write the augmented matrix corresponding to the following systems of equations.
(a)
x + y
= 1
2x
− 3z = 0
− 2y + 2z = 5
(b)
x + y = 1
2x − 3y = 4
(c)
3x + 2y − 4z = 2
x − y + 3z = 1
−x + y + 4z = 0
(10) Solve the system of equations using any applicable method. If the system is inconsistent, show
this.
(a)
x + y
= 2
2x
− 3z = −8
− 2y + 2z = −2
(b)
x + y + z = 5
2x − y + 5z = 1
y − z = 3
(c)
x − 3y = 4
3x + 2y = −5
(11) Solve the systems of equations by using Crammer’s rule if possible. If Crammer’s rule is not
applicable, state this and explain why.
(a)
2x − 4y = 5
2x − 3y = 4
(b)
x + 7y = 2
2x − 3y = 7
(c)
6x + 9y = 0
4x + 6y = −6

−3

(12) Consider the matrix B =  2
1
undefined, explain why.
(a) The entry b22
(b) The minor of b31
(c) The cofactor of b31
(d) The entry b24
1
2
0

4

−1 . Find each of the following if it exists. If it is
5
(e) The minor of b42
(f) The cofactor of b12
(13) Find the inverse of each matrix if it exists. If it doesn’t exists, show this.
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1
1
4
6
3
(a) A =
,
(b) B =
,
(c) C =
7
−2
6
9
1
(14) Use a matrix inverse to solve the system of equations if possible.
(a)
−2x − 2y = 9
−x + 2y = −3
(b)
−2x + 4y = 5
x − 4y = −3
−2
5
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