Chapter 10: Q .32. (page 812)

In Exercises 32–36,
(a) compute $u . v$
(b) find the angle between u and v, and
(c) find $p r o j_{u} v$

Short Answer

Expert verified

Part a) $u . v = 37$

Part b) angle is $θ = \cos^{- 1} (\frac{37}{\sqrt{1378}})$

Part c) $The value of {proj}_{u} v is \frac{37}{26} ⟨ 1, 5 ⟩ .$

Step by step solution

Step 1:Given information Part a)

u and v are vectors

Part a) Simplification

$Consider the vector u = ⟨ 1, 5 ⟩, v = ⟨ 2, 7 ⟩ .$

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

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

$u \cdot v = 1 (2) + 5 (7)$

$= 37$

Step 3:Part b) Given information

$u, v$ are given vectors

Step 4:Part b)Simplification

$u . v = 37$

$Also, u = ⟨ 1, 5 ⟩$

Therefore

$‖ u ‖ = \sqrt{1^{2} + 5^{2}}$

$= \sqrt{26}$

$Also, v = ⟨ 2, 7 ⟩$

$‖ v ‖ = \sqrt{2^{2} + 7^{2}}$

$= \sqrt{53}$

now,

$\cos θ = \frac{37}{\sqrt{26} \sqrt{53}}$

$\Rightarrow θ = \cos^{- 1} (\frac{37}{\sqrt{1378}})$

Hence, the angle between the vectors is $\cos^{- 1} (\frac{37}{\sqrt{1378}})$

Part c):Given information

$the vector u = ⟨ 1, 5 ⟩, v = ⟨ 2, 7 ⟩$

Step 6:Part c) Simplification

$\Rightarrow u . v = 37$ calculated

and

$‖ u ‖ = \sqrt{26}$

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

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

${proj}_{u} v = \frac{37}{(\sqrt{26})^{2}} ⟨ 1, 5 ⟩$

$= \frac{37}{26} ⟨ 1, 5 ⟩$

$Hence, the value of {proj}_{u} v is \frac{37}{26} ⟨ 1, 5 ⟩ .$

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In Exercises 32–36,
(a) compute $u . v$
(b) find the angle between u and v, and
(c) find $p r o j_{u} v$

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) computeu.v(b) find the angle between u and v, and(c) findprojuv

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

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Decision Maths

Pure Maths

Statistics

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 $p r o j_{u} v$