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Dot Product & Projections SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 11.3 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: • Know how to compute the dot product of two vectors. • Be able to use the dot product to find the angle between two vectors; and, in particular, be able to determine if two vectors are orthogonal. • Know how to compute the direction cosines of a vector. • Be able to decompose vectors into orthogonal components. And, know how to compute the orthogonal projection of one vector onto another. PRACTICE PROBLEMS: − − 1. For each of the following, compute → u ·→ v based on the given information. − − (a) → u = h3, −1i; → v = h2, −5i 11 → − − (b) u = h4, −5, 1i; → v = h3, 6, −1i −19 − − (c) → u = i − 3j + 4k; → v = 9i − 2j; 15 π − − − − (d) k→ u k = 3; k→ v k = 4; the angle between → u and → v is 4 √ 6 2 2. Explain why the operation (u · v) · w does not make sense. u · v is a scalar. We cannot take the dot product of a scalar with a vector. − − 3. Determine whether the angle between → v = h1, 2, 3i and → w = h−6, 4, −1i is acute, obtuse, or right. Explain. − − Since → v ·→ w = −1 < 0, the angle between the two vectors is obtuse. − − 4. Give an example of a vector which is orthogonal to both → v = h1, 1, 1i and → w = h2, 0, 4i. (Hint: Set up a system of equations.) − Any scalar multiple of → n = h−2, 1, 1i is orthogonal to both given vectors. 1 5. Consider the triangle with vertices a, b, and c. Use vectors to compute the angle between the diagonal which extends from vertex a to vertex b and the line segment which extends from vertex a to vertex c. (Verify your answer with HW 11.1 #3c.) √ ! 13 cos−1 √ 14 6. Consider the triangle, shown below,with vertices A(1, −2, 6), B(3, 0, −1), and C(−2, 1, 0). Compute all three angles of the triangle. 2 42 √ √ Angle A has a measure of cos radians. 57 54 15 Angle B has a measure of cos−1 √ √ radians. 57 27 12 Angle C has a measure of cos−1 √ √ radians. 54 27 (NOTE: Once you find any two of the angles, you can use the fact that the sum of all of the angles must be π radians to compute the remaining angle.) −1 → − − 7. Let → v = h1, 2i and b = h−3, 4i. → − − − (a) Find the vector component of → v along b and the vector component of → v or→ − thogonal to b . → − 3 4 → − → − → v = The vector component of v along b is Proj− and the vector − , b 55 → − 8 6 − − → − → v = , component of → v orthogonal to b is → v − Proj− . b 5 5 → − − (b) Sketch → v , b , and the vector components that you found in part (a). 8. Express v = i + 2j + 3k as the sum of a vector parallel to b = −2i + 4j − k and a vector perpendicular to b 2 4 1 9 10 22 v = − , ,− + , , ; Detailed Solution: Here 7 7 7 7 7 7 9. Suppose that u 6= 0 and v 6= 0. Under what condition will ku + vk = ku − vk? Explain. The result follows if u is orthogonal to v; Detailed Solution: Here 3 10. The following questions deal with finding the distance from a point to a line: (a) Given three points A, B, and P in 2-space or 3-space as shown in the picture below, describe two different ways that you could use the dot product to calculate the distance, d, between the point P and the line which contains A and B. OPTION 1: −→ −→ → AP . Then, use Pythagorean Theorem to One can compute AP and Proj− AB calculate d. OPTION 2: −→ −→ −→ − → − → One can compute ProjAB AP . Then, d = AP − ProjAB AP . (b) Use one of your methods from part (a) to compute the distance from the point P (5, 3, 0) to the line containing A(1, 0, 1) and B(2, 3, 1). r 91 ; Video Solution: Here d= 10 11. Consider the triangle shown below which is formed by vectors v and w. Prove Pythagorean’s Theorem kv + wk2 = kvk2 + kwk2 . (Hint: Use properties of the dot product to expand kv + wk2 .) 4 We expand kv + wk2 using properties of the dot product: kv + wk2 = (v + w) · (v + w) =v·v+v·w+w·v+w·w = kvk2 + 2(v · w) + kwk2 (since v · w = w · v) = kvk2 + kwk2 (since v ⊥ w ⇔ v · w = 0) Thus, kv + wk2 = kvk2 + kwk2 , as promised. 12. Let A and B be endpoints of a diameter of a circle with a radius of r . And, suppose that C is any other point on the circle, as shown below. −→ −−→ Prove that triangle ABC is a right triangle. (Hint: Express each of CA and CB as −→ the combination of CO and some other vector.) Notice that CA = CO + OA and CB = CO + OB. But, OB = −OA. Thus, CB = CO − OA. We will show that CA ⊥ CB by showing that CA · CB = 0. CA · CB = (CO + OA) · (CO − OA) = CO · CO − CO · OA + OA · CO − OA · OA = kCOk2 − kOAk2 = r2 − r2 =0 So, CA ⊥ CB and the triangle is a right triangle. 5 − − 13. Let → v and → w be vectors, either both in R2 or in R3 . Prove the Cauchy-Schwarz − − − − Inequality: |→ v ·→ w | ≤ k→ v kk→ w k. − − Suppose that θ is the angle between → v and → w . Then: − − − − |→ v ·→ w | = |k→ v kk→ w k cos θ| → − − = (k v k) (k→ w k) (| cos θ|) → − → − ≤ (k v k) (k w k) (1) − − = k→ v kk→ wk − − − − Thus, |→ v ·→ w | ≤ k→ v kk→ w k, as promised. − 14. Let → v = h1, 2, 3i. − (a) Compute the direction cosines of → v. − Let α, β, and γ be the angles between → v and i, j, and k, respectively. Then: 1 2 3 cos α = √ , cos β = √ , and cos γ = √ 14 14 14 (b) Verify that cos2 α + cos2 β + cos2 γ = 1, where α, β, and γ be the angles between → − v and i, j, and k, respectively. Using the information from part (a), we obtain: 2 2 2 cos α + cos β + cos γ = 1 √ 14 2 + 1 4 9 + + 14 14 14 =1 = 6 2 √ 14 2 + 3 √ 14 2