Chapter 10: Q 27. (page 812)

In Exercises 24-27, find ${comp}_{u} v, {proj}_{u} v$ and the component of v orthogonal tou.
role="math" localid="1649693816584" $u = <3, 1, - 2>, v = <- 6, - 2, 4>$

Short Answer

Expert verified

The values are ${comp}_{u} v = - 2 \sqrt{14}, {proj}_{u} v = <- 6, - 2, 4>$ and the component of v orthogonal tou is $<0, 0, 0>$ .

Step by step solution

Step 1. Given information.

The given vectors are:

$u = <3, 1, - 2>, v = <- 6, - 2, 4>$

Step 2. Find the component of the projection of v onto u.

${comp}_{u} v = \frac{u \cdot v}{||u||} u = <3, 1, - 2> and v = <- 6, - 2, 4> u \cdot v = 3 (- 6) + 1 (- 2) - 2 (4) = - 18 - 2 - 8 = - 28 ||u|| = \sqrt{3^{2} + 1^{2} + {(- 2)}^{2}} = \sqrt{9 + 1 + 4} = \sqrt{14}$

Then,

${comp}_{u} v = \frac{u \cdot v}{||u||} = \frac{- 28}{\sqrt{14}} = \frac{- 28}{\sqrt{14}} \times \frac{\sqrt{14}}{\sqrt{14}} = - 2 \sqrt{14}$

Therefore, role="math" localid="1649694131877" ${comp}_{u} v = - 2 \sqrt{14}$ .

Step 3. Find the vector projection of v onto u.

${proj}_{u} v = \frac{u \cdot v}{{||u||}^{2}} u = \frac{- 28}{{(\sqrt{14})}^{2}} <3, 1, - 2> = - 2 <3, 1, - 2> = <- 6, - 2, 4>$

Therefore, the vector projection of v onto uis $<- 6, - 2, 4>$ .

Step 4. Find the component of v orthogonal to u.

The component of v orthogonal tou is $v - {proj}_{u} v$ .

$v - {proj}_{u} v = <- 6, - 2, - 4> - <- 6, - 2, - 4> = <0, 0, 0>$

Therefore, the component of v orthogonal to u is $<0, 0, 0>$ .

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In Exercises 24-27, find ${comp}_{u} v, {proj}_{u} v$ and the component of v orthogonal tou.
role="math" localid="1649693816584" $u = <3, 1, - 2>, v = <- 6, - 2, 4>$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the component of the projection of v onto u.

Step 3. Find the vector projection of v onto u.

Step 4. Find the component of v orthogonal to u.

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In Exercises 24-27, find compuv,projuv and the component of v orthogonal tou.role="math" localid="1649693816584" u=3,1,-2,v=-6,-2,4

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the component of the projection of v onto u.

Step 3. Find the vector projection of v onto u.

Step 4. Find the component of v orthogonal to u.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Discrete Mathematics

Mechanics Maths

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

In Exercises 24-27, find ${comp}_{u} v, {proj}_{u} v$ and the component of v orthogonal tou.
role="math" localid="1649693816584" $u = <3, 1, - 2>, v = <- 6, - 2, 4>$