Chapter 10: Q .33. (page 812)

In Exercises 32-36,
(a) compute u . v
(b) find the angle between u and v, and
(c) find ${proj}_{u} v$

Short Answer

Expert verified

Part a) $u . v = - 1$

Part b)The angle between the vectors is $θ = \cos^{- 1} (\frac{- 1}{2 \sqrt{2451}})$

Part c) ${Value ofproj}_{n} v is \frac{- 1}{38} (- 2, 3, 5) .$

Step by step solution

Step 1:Given information Part a)

u and v are vectors

Part a) Simplification

$Consider the vector u = ⟨ - 2, 3, 5 ⟩, v = ⟨ 13, - 5, 8 ⟩$

$If u and v are the vector such that u = <u_{1}, v_{1}, w_{1}> and v = <u_{2}, v_{2}, w_{2}>, then dot product is$

$given by u . v = u_{1} v_{1} + u_{2} v_{2} + w_{1} w_{2}$

$u \cdot v = - 2 (13) + 3 (- 5) + 5 (8)$

$= - 26 - 15 + 40$ $= - 1$

Step 3:Part b) Given information

$the vector u = ⟨ - 2, 3, 5 ⟩, v = ⟨ 13, - 5, 8 ⟩$

Step 4:Part b)Simplification

$u . v = - 1$

$Also, u = ⟨ - 2, 3, 5 ⟩$

$‖ u ‖ = \sqrt{(- 2)^{2} + 3^{2} + 5^{2}}$

$= \sqrt{38}$

$Also, v = ⟨ 13, - 5, 8 ⟩$

$‖ v ‖ = \sqrt{13^{2} + (- 5)^{2} + 8^{2}}$

$\sqrt{258}$

now

$\cos θ = \frac{- 1}{\sqrt{38} \sqrt{358}}$

$= \frac{- 1}{2 \sqrt{2451}}$

Therefore

$θ = \cos^{- 1} (\frac{- 1}{2 \sqrt{2451}})$

Hence, the angle between the vectors is $θ = \cos^{- 1} (\frac{- 1}{2 \sqrt{2451}})$

Part c):Given information

$the vector u = ⟨ - 2, 3, 5 ⟩, v = ⟨ 13, - 5, 8 ⟩$

Step 6:Part c) Simplification

$u . v = - 1$ (calculated)

$‖ u ‖ = \sqrt{38}$ (calculated)

$Let u be any non-zero vector, then the vector projection of v onto u is given by$

${proj}_{u} v = \frac{u \cdot v}{‖ u ‖^{2}} u .Now, substitute the values in {proj}_{u} v = \frac{u \cdot v}{‖ u ‖^{2}} u$

${proj}_{u} v = \frac{- 1}{(\sqrt{38})^{2}} ⟨ - 2, 35 ⟩$

$= \frac{- 1}{38} ⟨ - 2, 3, 5 ⟩$

${Hence, value ofproj}_{u} v is \frac{- 1}{38} (- 2, 3, 5) .$

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In Exercises 32-36,
(a) compute u . v
(b) find the angle between u and v, and
(c) find ${proj}_{u} v$

Short Answer

Step by step solution

Step 1:Given information Part a)

Part a) Simplification

Step 3:Part b) Given information

Step 4:Part b)Simplification

Part c):Given information

Step 6:Part c) Simplification

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In Exercises 32-36,(a) compute u . v(b) find the angle between u and v, and(c) find projuv

Short Answer

Step by step solution

Step 1:Given information Part a)

Part a) Simplification

Step 3:Part b) Given information

Step 4:Part b)Simplification

Part c):Given information

Step 6:Part c) Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

In Exercises 32-36,
(a) compute u . v
(b) find the angle between u and v, and
(c) find ${proj}_{u} v$