Linear Algebra/Length and Angle Measures/Solutions

From Wikibooks, open books for an open world
< Linear Algebra‎ | Length and Angle Measures
Jump to navigation Jump to search

Solutions[edit]

This exercise is recommended for all readers.
Problem 1

Find the length of each vector.

Answer
This exercise is recommended for all readers.
Problem 2

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

Answer
  1. 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

Linalg angle bisection.svg

(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.

References[edit]

  • 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