Linear Algebra/Length and Angle Measures/Solutions

SolutionsEdit

This exercise is recommended for all readers.
Problem 1

Find the length of each vector.

  1.  
  2.  
  3.  
  4.  
  5.  
Answer
  1.  
  2.  
  3.  
  4.  
  5.  
This exercise is recommended for all readers.
Problem 2

Find the angle between each two, if it is defined.

  1.  
  2.  
  3.  
Answer
  1.  
  2.  
  3. Not defined.
This exercise is recommended for all readers.
Problem 3

During maneuvers preceding the Battle of Jutland, the British battle cruiser Lion moved as follows (in nautical miles):   miles north,   miles   degrees east of south,   miles at   degrees east of north, and   miles at   degrees east of north. Find the distance between starting and ending positions (O'Hanian 1985).

Answer

We express each displacement as a vector (rounded to one decimal place because that's the accuracy of the problem's statement) and add to find the total displacement (ignoring the curvature of the earth).

 

The distance is  .

Problem 4

Find   so that these two vectors are perpendicular.

 
Answer

Solve   to get  .

Problem 5

Describe the set of vectors in   orthogonal to this one.

 
Answer

The set

 

can also be described with parameters in this way.

 
This exercise is recommended for all readers.
Problem 6
  1. Find the angle between the diagonal of the unit square in   and one of the axes.
  2. Find the angle between the diagonal of the unit cube in   and one of the axes.
  3. Find the angle between the diagonal of the unit cube in   and one of the axes.
  4. What is the limit, as   goes to  , of the angle between the diagonal of the unit cube in   and one of the axes?
Answer
  1. We can use the  -axis.
     
  2. Again, use the  -axis.
     
  3. The  -axis worked before and it will work again.
     
  4. Using the formula from the prior item,  .
Problem 7

Is any vector perpendicular to itself?

Answer

Clearly   is zero if and only if each   is zero. So only   is perpendicular to itself.

This exercise is recommended for all readers.
Problem 8

Describe the algebraic properties of dot product.

  1. Is it right-distributive over addition:  ?
  2. Is is left-distributive (over addition)?
  3. Does it commute?
  4. Associate?
  5. How does it interact with scalar multiplication?

As always, any assertion must be backed by either a proof or an example.

Answer

Assume that   have components  .

  1. Dot product is right-distributive.
     
  2. Dot product is also left distributive:  . The proof is just like the prior one.
  3. Dot product commutes.
     
  4. Because   is a scalar, not a vector, the expression   makes no sense; the dot product of a scalar and a vector is not defined.
  5. This is a vague question so it has many answers. Some are (1)   and  , (2)   (in general; an example is easy to produce), and (3)   (the connection between norm and dot product is that the square of the norm is the dot product of a vector with itself).
Problem 9

Verify the equality condition in Corollary 2.6, the Cauchy-Schwartz Inequality.

  1. Show that if   is a negative scalar multiple of   then   and   are less than or equal to zero.
  2. Show that   if and only if one vector is a scalar multiple of the other.
Answer
  1. Verifying that   for   and   is easy. Now, for   and  , if   then  , which is   times a nonnegative real. The   half is similar (actually, taking the   in this paragraph to be the reciprocal of the   above gives that we need only worry about the   case).
  2. We first consider the   case. From the Triangle Inequality we know that   if and only if one vector is a nonnegative scalar multiple of the other. But that's all we need because the first part of this exercise shows that, in a context where the dot product of the two vectors is positive, the two statements "one vector is a scalar multiple of the other" and "one vector is a nonnegative scalar multiple of the other", are equivalent. We finish by considering the   case. Because   and  , we have that  . Now the prior paragraph applies to give that one of the two vectors   and   is a scalar multiple of the other. But that's equivalent to the assertion that one of the two vectors   and   is a scalar multiple of the other, as desired.
Problem 10

Suppose that   and  . Must  ?

Answer

No. These give an example.

 
This exercise is recommended for all readers.
Problem 11

Does any vector have length zero except a zero vector? (If "yes", produce an example. If "no", prove it.)

Answer

We prove that a vector has length zero if and only if all its components are zero.

Let   have components  . Recall that the square of any real number is greater than or equal to zero, with equality only when that real is zero. Thus   is a sum of numbers greater than or equal to zero, and so is itself greater than or equal to zero, with equality if and only if each   is zero. Hence   if and only if all the components of   are zero.

This exercise is recommended for all readers.
Problem 12

Find the midpoint of the line segment connecting   with   in  . Generalize to  .

Answer

We can easily check that

 

is on the line connecting the two, and is equidistant from both. The generalization is obvious.

Problem 13

Show that if   then   has length one. What if  ?

Answer

Assume that   has components  . If   then we have this.

 
 

If   then   is not defined.

Problem 14

Show that if   then   is   times as long as  . What if  ?

Answer

For the first question, assume that   and  , take the root, and factor.

 

For the second question, the result is   times as long, but it points in the opposite direction in that  .

This exercise is recommended for all readers.
Problem 15

A vector   of length one is a unit vector. Show that the dot product of two unit vectors has absolute value less than or equal to one. Can "less than" happen? Can "equal to"?

Answer

Assume that   both have length  . Apply Cauchy-Schwartz:  .

To see that "less than" can happen, in   take

 

and note that  . For "equal to", note that  .

Problem 16

Prove that  

Answer

Write

 

and then this computation works.

 
Problem 17

Show that if   for every   then  .

Answer

We will prove this demonstrating that the contrapositive statement holds: if   then there is a   with  .

Assume that  . If   then it has a nonzero component, say the  -th one  . But the vector   that is all zeroes except for a one in component   gives  . (A slicker proof just considers  .)

Problem 18

Is  ? If it is true then it would generalize the Triangle Inequality.

Answer

Yes; we can prove this by induction.

Assume that the vectors are in some  . Clearly the statement applies to one vector. The Triangle Inequality is this statement applied to two vectors. For an inductive step assume the statement is true for   or fewer vectors. Then this

 

follows by the Triangle Inequality for two vectors. Now the inductive hypothesis, applied to the first summand on the right, gives that as less than or equal to  .

Problem 19

What is the ratio between the sides in the Cauchy-Schwartz inequality?

Answer

By definition

 

where   is the angle between the vectors. Thus the ratio is  .

Problem 20

Why is the zero vector defined to be perpendicular to every vector?

Answer

So that the statement "vectors are orthogonal iff their dot product is zero" has no exceptions.

Problem 21

Describe the angle between two vectors in  .

Answer

The angle between   and   is found (for  ) with

 

If   or   is zero then the angle is   radians. Otherwise, if   and   are of opposite signs then the angle is   radians, else the angle is zero radians.

Problem 22

Give a simple necessary and sufficient condition to determine whether the angle between two vectors is acute, right, or obtuse.

Answer

The angle between   and   is acute if  , is right if  , and is obtuse if  . That's because, in the formula for the angle, the denominator is never negative.

This exercise is recommended for all readers.
Problem 23

Generalize to   the converse of the Pythagorean Theorem, that if   and   are perpendicular then  .

Answer

Suppose that  . If   and   are perpendicular then

 

(the third equality holds because  ).

Problem 24

Show that   if and only if   and   are perpendicular. Give an example in  .

Answer

Where  , the vectors   and   are perpendicular if and only if  , which shows that those two are perpendicular if and only if  . That holds if and only if  .

Problem 25

Show that if a vector is perpendicular to each of two others then it is perpendicular to each vector in the plane they generate. (Remark. They could generate a degenerate plane— a line or a point— but the statement remains true.)

Answer

Suppose   is perpendicular to both   and  . Then, for any   we have this.

 
Problem 26

Prove that, where   are nonzero vectors, the vector

 

bisects the angle between them. Illustrate in  .

Answer

We will show something more general: if   for  , then   bisects the angle between   and  

 

(we ignore the case where   and   are the zero vector).

The   case is easy. For the rest, by the definition of angle, we will be done if we show this.

 

But distributing inside each expression gives

 

and  , so the two are equal.

Problem 27

Verify that the definition of angle is dimensionally correct: (1) if   then the cosine of the angle between   and   equals the cosine of the angle between   and  , and (2) if   then the cosine of the angle between   and   is the negative of the cosine of the angle between   and  .

Answer

We can show the two statements together. Let  , write

 

and calculate.

 
This exercise is recommended for all readers.
Problem 28

Show that the inner product operation is linear: for   and  ,  .

Answer

Let

 

and then

 

as required.

This exercise is recommended for all readers.
Problem 29

The geometric mean of two positive reals   is  . It is analogous to the arithmetic mean  . Use the Cauchy-Schwartz inequality to show that the geometric mean of any   is less than or equal to the arithmetic mean.

Answer

For  , set

 

so that the Cauchy-Schwartz inequality asserts that (after squaring)

 

as desired.

? Problem 30

A ship is sailing with speed and direction  ; the wind blows apparently (judging by the vane on the mast) in the direction of a vector  ; on changing the direction and speed of the ship from   to   the apparent wind is in the direction of a vector  .

Find the vector velocity of the wind (Ivanoff & Esty 1933).

Answer

This is how the answer was given in the cited source.

The actual velocity   of the wind is the sum of the ship's velocity and the apparent velocity of the wind. Without loss of generality we may assume   and   to be unit vectors, and may write

 

where   and   are undetermined scalars. Take the dot product first by   and then by   to obtain

 

Multiply the second by  , subtract the result from the first, and find

 

Substituting in the original displayed equation, we get

 
Problem 31

Verify the Cauchy-Schwartz inequality by first proving Lagrange's identity:

 

and then noting that the final term is positive. (Recall the meaning

 

and

 

of the   notation.) This result is an improvement over Cauchy-Schwartz because it gives a formula for the difference between the two sides. Interpret that difference in  .

Answer

We use induction on  .

In the   base case the identity reduces to

 

and clearly holds.

For the inductive step assume that the formula holds for the  , ...,   cases. We will show that it then holds in the   case. Start with the right-hand side

 
 

and apply the inductive hypothesis

 

to derive the left-hand side.

ReferencesEdit

  • O'Hanian, Hans (1985), Physics, 1, W. W. Norton 
  • Ivanoff, V. F. (proposer); Esty, T. C. (solver) (Feb. 1933), "Problem 3529", American Mathematical Mothly 39 (2): 118