Chapter 10: Q 24. (page 812)

In Exercises 24-27, find comp_uv, proj_uv, and the component of v orthogonal tou.
$u = <1, 2>, v = <3, 5>$

Short Answer

Expert verified

The values are ${comp}_{u} v = \frac{13}{\sqrt{5}}, {proj}_{u} v = \frac{13}{5} <1, 2>$ and the component of v orthogonal tou is $<\frac{2}{5}, \frac{- 1}{5}>$ .

Step by step solution

Step 1. Given information.

The given vectors are:

$u = <1, 2> and v = <3, 5>$

Step 2. Find compu v.

${comp}_{u} v = \frac{u \cdot v}{||u||}$ is the projection of vonto u.

$u = <1, 2> and v = <3, 5> u \cdot v = 1 (3) + 2 (5) = 3 + 10 = 13 ||u|| = \sqrt{1^{2} + 2^{2}} = \sqrt{1 + 4} = \sqrt{5}$

then, localid="1649668447466" ${comp}_{u} v = \frac{u \cdot v}{||u||}$ is:

localid="1649667564938" $= \frac{13}{\sqrt{5}}$

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

${proj}_{u} v = \frac{u \cdot v}{{||u||}^{2}} u = \frac{13}{{(\sqrt{5})}^{2}} <1, 2> = \frac{13}{5} <1, 2>$

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

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

$v - {proj}_{u} v = <3, 5> - \frac{13}{5} <1, 2> = <3, 5> - <\frac{13}{5}, \frac{26}{5}> = <\frac{2}{5}, \frac{- 1}{5}>$

Therefore, the component of v orthogonal to u is localid="1649668403950" $<\frac{2}{5}, \frac{- 1}{5}>$ .

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In Exercises 24-27, find comp_uv, proj_uv, and the component of v orthogonal tou.
$u = <1, 2>, v = <3, 5>$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find compu v.

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.u=1,2,v=3,5

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find compu v.

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

Applied Mathematics

Theoretical and Mathematical Physics

Decision Maths

Statistics

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Exercises 24-27, find comp_uv, proj_uv, and the component of v orthogonal tou.
$u = <1, 2>, v = <3, 5>$