# Solutions to Problem Set 1

Math 311-01/02

1-25-2018

[Euclidean space]

1. Plot the points , , and .

2. Find the exact distance between and .

3. Find the exact distance between and .

[Vectors]

4. Suppose

Find:

(a) .

(b)

(c)

(a) .

(b)

(c)

5. Suppose

Find:

(a) .

(b)

(c)

(a) .

(b)

(c)

6. Compute

7. Suppose .

(a) Find .

(b) Find a unit vector in the same direction as .

(c) Find a unit vector in the opposite direction to .

(a) .

(b) .

(c) .

8. Find a vector with length 6 and the same direction as .

Therefore, is a unit vector with the same direction as . Hence, is a vector with length 6 and the same direction as .

9. (a) For the points and , find .

(b) Find a unit vector in the opposite direction to .

(a) .

(b) , so is a unit vector in the opposite direction to .

10. Prove that the vectors and are parallel.

Hence, the vectors are parallel.

11. Prove that the vectors and are NOT parallel.

Suppose

Then

This gives

The first equation gives , but the second gives . This contradiction shows there is no such k. Hence, the vectors aren't parallel.

Life shrinks or expands in proportion to one's courage. - Anaïs Nin

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