Vectors and geometry



Vectors: practice and review

Skills and Concepts:

• Familiarity and facility with both ways we learned to represent a vector: component form and with magnitude and direction, including converting between the two representations.

• Calculations (addition, subtraction, scalar multiplication, and dot product) shown visually (except dot product) as well as arithmetically

• Finding unit vectors in the same direction as a given vector

• Determining if two vectors are parallel, perpendicular or neither (2 ways to show parallelism)

• Finding the angle between two vectors using a vector method and applying this to problems

• Writing vector proofs

These practice and review problems will help prepare you for a quiz tomorrow on vectors.

So that you get practice using all the ideas that you need to review, be sure to follow the directions about doing problems algebraically vs. visually/graphically.

1. Consider the vectors v = and w = .

a. Calculate v – w algebraically, then make a drawing on graph paper that confirms your answer.

b. Find 3v + 1.5w visually on graph paper, then algebraically confirm your answer.

c. Calculate v ∙ w.

d. Are vectors v and w parallel? Justify your answer.

e. Are vectors v and w orthogonal? Justify your answer.

f. Find the angle measure between vectors v and w.

g. Find the angle measure between vector v and the x-axis.

2. This problem will lead you through a vector proof of this geometry theorem:

“The diagonals of a rhombus must be perpendicular.”

Suppose that the given diagram shows a rhombus, with v = [pic] and w = [pic].

a. What is a rhombus?

b. Is it true that v = w ? Explain why or why not.

c. Is it true that | v | = | w | ? Explain why or why not.

d. What does [pic] equal, in terms of v and w?

e. What does [pic] equal, in terms of v and w?

f. Calculate the dot product [pic] ∙ [pic]. Simplify your answer as much as possible.

g. Prove that [pic] and [pic] are perpendicular.

3. This problem will lead you through a vector proof of this geometry theorem: “If a parallelogram has diagonals of equal length, then it must be a rectangle.”

Suppose that the given diagram shows a parallelogram with diagonals of equal length. Let v = [pic] and w = [pic].

a. What is the given knowledge about arrows [pic] and [pic]?

b. Identify each of the following in terms of v and/or w.

[pic] =

[pic] =

c. Prove that (v + w) ∙ (v + w) = (v – w) ∙ (v – w).

d. Prove that v ∙ w = 0.

e. Explain why the parallelogram must be a rectangle.

4. Suppose that v = 7i – 2j. Let w be a unit vector in the same direction as v.

a. Find w in component form.

b. Calculate these dot products: v ∙ v, v ∙ w, and w ∙ w.

c. Find | v + w | without finding v + w first.

Hint: Use the relationship between magnitude and dot product.

Return to the book and try even problems within the problem ranges I assigned to practice the skills not included in this review to prepare for the quiz (as needed).

One more thing

the character Vector from the movie “Despicable Me”: “I go by the name of Vector. It’s a mathematical term, represented by an arrow with both direction and magnitude. Vector! That’s me, because I’m committing crimes with both direction and magnitude. Oh yeah!”

59 seconds at

Answers

1. a. b.

c. –32 d. No, because their slopes are unequal: –5/2 and –4/6.

e. No, because v ∙ w ≠ 0. f. Use the formula from page 468: [pic]≈ 145.49°.

g. Draw a picture of this vector, then use right triangle trigonometry. You could have chosen either an acute angle or an obtuse angle so there are two acceptable answers:

tan–1([pic]) ≈ 68.20° or its supplementary angle 111.80°.

2. a. “quadrilateral with all four side lengths equal” or “parallelogram with two adjacent side lengths equal”

b. No, because the vectors don’t have the same run and rise.

c. Yes, because sides of a rhombus have equal lengths. d. w – v e. v + w

f. (w – v)·(v + w) = w · v + w · w – v · v – v · w

= | w |2 – | v |2

= 0 because | v | and | w | are equal

g. Since the vectors’ dot product equals 0, the vectors are perpendicular.

3. a. [pic] (the magnitudes of BD and AC are equal)

b. [pic] = v + w and [pic] = v – w

c. (v + w)((v+w) = |v + w|2 = |[pic]|2 = |[pic]|2 = |v – w|2 = (v – w)((v – w).

d. Since we proved that (v+w)((v+w) = (v-w)((v-w), we can now expand these expressions to show that v(v + 2v(w + w(w = v(v – 2v(w + w(w. Simplifying this we get 2v(w = –2v(w, which leads to 4v(w= 0, so v(w= 0.

e. Since the vectors v and w have a dot product equal to zero, they are orthogonal vectors. Parallelograms with one right angle is a rectangle.

4. a. [pic] b. 53, [pic], 1

c. First find the square of the desired length, using dot product:

| v + w |2 = (v + w)·(v + w) = v · v + v · w + w · v + w · w

= 53 + [pic] + [pic] + 1 ≈ 68.560

Then | v + w | ≈ [pic]≈ 8.280.

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