Homework Assignment 4
CSC 170 Numerical Methods
Student Name
Section
Instructor
Due Date
Exercise 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Points
Score
4 4
4 4 4
4
4 4
4 4
4 4
4 4 4
4
4 4
4 4
4 4
Total
4 4 4 100 points
Textbook Reading Assignment
Thoroughly review your course lecture notes for the topic(s) given below.
Vectors
Use Mathematica / WolframAlpha to solve each of these.
Show all necessary work to answer each of the following.
(1)
Two vectors v and w are said to be parallel if there is a nonzero scalar  such that
v =  w . In this case, the angle  between v and w is 0 or  .
Prove that the vectors v = i − 3 j and w = 2 i − 6 j are parallel.
Hint: Define v = { a , b } and w = { a , b } . Use Mathematica to show that one vector
is a scalar multiple of the other.
(2)
Determine whether the vectors v = 16 i − 8 j and w = 4 i − 2 j are parallel.
Hint: The scalar product of two, three - dimensional vectors is: { a , b , c } . { x , y , z } .
(3)
Two vectors v and w are said to be orthogonal if and only if their dot product v  w is
0 . Determine whether the vectors v = 4 i + 3 j and w = 3 i − 4 j are orthogonal.
Hint: The scalar or Dot product of two, three - dimensional vectors is:
{ a, b, c } . { x, y, z }.
(4)
Find a such that the vectors v = 2 i − a j and w = i + 3 j are orthogonal.
Hint: Use Mathematica to verify your result.
(5)
Determining the magnitude || v || of a vector, is similar to using the Theorem of
Pythagoras. In three - dimensional space, || v || = ( v 1
2
+ v2
2
2 1/2
+ v3 )
.
Find the magnitude || v || of the vector v = 4 i − j + 3 k . Express your answer as
a radical in simplified radical form.
Hint: To find the magnitude of vector { x , y , z } use Norm[ { x , y , z } ] .
(6)
Find the magnitude || u || of the vector u = 5 i + 0 j + 4 k . Express your answer as
a radical in simplified radical form.
Hint: To find the magnitude of vector { x , y , z } use Norm[ { x , y , z } ] .
(7)
If || v || = 4 , what is || − 9 v || ?
Hint: Construct a vectors whose " Norm " is 4 and then find the length of − 9 v .
(8)
Which of these is NOT a unit vector in three - dimensional space?
(a) ( 1 , 0 , 0 ) (b) ( 1 , 0 , 1 ) (c) ( 0 , 1 , 0 ) (d) ( 0 , 0 , 1 ) (e) Both (a) and (d)
Hint: To find the magnitude of vector { x , y , z } use Norm[ { x , y , z } ] . The magnitude
of a unit vector must equal a length of 1 . Enter your answer in your Mathematica
Notebook file using comments (* *) .
© Copyright 2015 by P . E . P .
1
CSC 170 Numerical Methods
Homework Assignment 4
Student Name
Section
(9)
The unit vector having the same direction as a given ( nonzero ) vector v is defined by
v / || v || . Find the unit vector having the same direction as v = i − 2 j .
Hint: Define a vector v = { a , b } and divide v by Norm[ v ] .
(10)
Find a vector v whose magnitude is 3 and whose component in the i direction is equal to
the component in the j direction.
Hint: Define a vector v = { a , b } such that a is equal to b and Norm[ v ] = 3 .
(11)
Consider the definition of the dot product of two vectors u and v .
u  v = || u || || v ||
cos 
where is  the angle between the vectors.
Find the angle between the vectors u = ( 3 , 0 ) and v = ( 0 , 3 ) .
Hint: Define a vector u = { a , b } and define a vector v = { x , y } . Set r to be
the ratio of the scalar product between a and b and the product of their magnitudes.
Compute ArcCos[ r ] .
(12)
Find the angle between the vectors u = ( 0 , 2 ) and v = ( 1 , 2 ) .
Hint: Refer to the above hint.
(13)
A vector u has a magnitude of 10 and an angle  = 120  that it makes with the
positive x - axis. Write the vector in the form a i + b j .
Hint: Define a vector u = { a , b } and define a vector v = { x , y } . Set r to be
the ratio of the scalar product between a and b and the product of their magnitudes.
Compute ArcCos[ r ] .
A woman pushes a wheelbarrow up an incline of 30 ° with a force of 50 pounds.
Express the force F as a vector in terms of i and j .
Hint: Consider the structure of the force that is given in the exercise below.
(14)
(15)
(16)
In elementary physics, the work W performed by a constant force F in moving an object
from point A to point B is defined as the magnitude of the force multiplied by the
distance traveled x or:
W = Fx
If F = 30 ( cos 45 ° i + sin 45 ° j ) pounds and x = 100 i feet, find the work W in
foot - pounds.
Hint: In Physics, " work " is defined as the vector scalar product between the applied
force and the distance traveled.
Find the work performed by a force of 3 pounds acting in the direction 60  to the
horizontal in moving an object 2 feet from ( 0 , 0 ) to ( 2 , 0 ) .
Hint: Consider the definition of " work " that is given in the exercise above.
(17)
Scalar multiplication refers to the multiplication of a vector by a constant  , producing a
vector in the __________ direction for  > 0 or __________ direction for   0 but of
different length.
(a)
same ; opposite
(b)
opposite ; opposite
(c)
same ; same
(d)
opposite ; same
Hint: Enter your answer in your Mathematica Notebook file using comments (* *) .
(18)
Vector addition is the operation of adding two or more vectors together into a vector
sum.
(a)
True
(b)
False
Hint: Enter your answer in your Mathematica Notebook file using comments (* *) .
© Copyright 2015 by P . E . P .
2
CSC 170 Numerical Methods
Homework Assignment 4
Student Name
(19)
Section
The vector cross product between vectors u and v is defined as:
u  v = (u2 v3 − u3 v2)i + (u3 v1 − u1 v3)j + (u1 v2 − u2 v1)k
Write, in a shorthand notation that takes the form of a determinant, the vector cross
product between vectors u and v .
Hint: The vector or Cross product of two, three - dimensional vectors in matrix form is:
MatrixForm [ { i , j , k } , { a , b , c } , { x , y , z } ] .
(20)
Consider the vectors u = 3 i + 4 j − 5 k and v = 2 i + k . Determine the vector
cross product between vectors u and v .
Hint: The vector or Cross product of two, three - dimensional vectors is:
Cross [ { a , b , c } , { x , y , z } ] .
(21)
Investigate the divergence of a vector. Is divergence a scalar or a vector?
Hint: Visit Mathematica Help and execute Div[{3x,2y,3z}] . Then execute the
command Div[{x,3y,3}] . In both cases, was a vector returned or a scalar?
(22)
Investigate the curl of a vector. Is curl a scalar or a vector?
Hint: Visit Mathematica or WolframAlpha Help and execute an example from the curl
calculator. Show the results of the execution of your example.
(23)
Investigate the WolframAlpha vector command, which has the form
vector {a, b, c} for integers a , b and c . Do this by using some random values
of a , b and c and observing the result.
(24)
Investigate a WolframAlpha normalize command, which has the form
normalize the vector (a, b, c) for real values a , b and c .
Do this by using some random values of a , b and c and observing the result.
(25)
Investigate the way in which WolframAlpha can perform a vector cross product.
Do this by running a command similar to the following: (4,1) x (-5,6)
© Copyright 2015 by P . E . P .
3