Calculus: The Language of Change

Exercise Set 15.4
Scalar Multiplication

solutions by Randall Childs and David Theobald

See the text for explicit instructions on using a Mathematica notebook to solve Exercise 1.

Where do these fancy graphs come from?

The graphs in this notebook were drawn with the aid of the special commands defined in several of the notebooks located under Chapter 15 in the MathmCTLC directory.  The Mathematica gymnastics used to define these special commands are too complicated to explain.  In fact, even the solutions author does not understand them.  Suffice it to say that they take a large set of simpler Mathematica graphics commands and combines them so that the user may draw useful pictures of vectors without crunching a lot of computer programming code.

Exercise 2

Compute the length of Y=(({{8}, {4}, {2}})) and the length of X=(({{-4}, {-2}, {-1}})).

Show that |Y|≠-2|X|.

If Y=cX, when is |Y|≠cX?

Y = (8,4,2)	;	X = (-4,-2,-1)

|Y| = (Sqrt[(8^2) + (4^2) + (2^2)]) = (Sqrt[84]) = 2 (Sqrt[21])

|X| = \(TraditionalForm
      \`\@\(\((\(-4\))\)\^2 + \((\(-2\))\)\^2 + \((\(-1\))\)\^2\)\) = (Sqrt[21])

|Y| = 2 |X|   ⇒   |Y| ≠ -2 |X|

Given Y = c X.  |Y| ≠ c |X| when X ≠ 0 and c < 0.

Exercise 3

Parallel Position Vectors

Use the geometric meaning of scalar multiplication to give an algebraic condition equivalent to two position vectors being parallel:

X is parallel to Y if and only if there exist a real number c such that c X = Y.

a)	X(4,5) and Y(6,7) are NOT parallel

(6/4) ≠ (7/5)

b)	X(4,5) and Y(8,10) are parallel

c = 2

c)	X(-3,4) and Y(6,-8) are parallel

c = -2

d)	X(3,4,5) and Y(6,7,8) are NOT parallel

(6/3) ≠ (7/4) ≠ (8/5)

e)	X(3,4,5) and Y(6,8,10) are parallel

c = 2

f)	X(-3,4,-5) and Y(6,-8,10) are parallel

c = -2

Exercise 4

A Line:

Describe geometrically the set of points that consists of the tips of all position vectors of the form (({{2  c}, {3  c}})), where c takes on all real number values.  Sketch several values, c=1, c=3/2, c=2, c=5/2, c=1/2, c=-1/2, c=-1, c=-3/2...  until you see the general pattern.

a)	The line in the plane containing the origin and (2,3)

Repeat part (a) using the 3-D position vector (({{3  c}, {5  c}, {8  c}})).

b)	The line in space containing the origin and (3,5,8)