Q35E Among all the unit vectors u鈫�=... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q35E (page 217)

Among all the unit vectors $\overset{鈫�}{u} = [\begin{matrix} x \\ y \\ z \end{matrix}]$ in $R^{3}$ , find the
one for which the sum role="math" localid="1659544842303" $x + 2 y + 3 z$ is minimal.

Short Answer

Expert verified

The vector with the minimal length $\overset{鈫�}{u} = [\begin{matrix} - 1 / \sqrt{14} \\ - 2 / \sqrt{14} \\ - 3 / \sqrt{14} \end{matrix}]$

Step by step solution

Step by step solution Step 1: Use Cauchy-Schwarz inequality

$| \overset{鈫�}{v} . \overset{鈫�}{w} | 猢� | | \overset{鈫�}{v} | | . | | \overset{鈫�}{w} | |$

Observe that $x + 2 y + 3 z = [\begin{matrix} x \\ y \\ z \end{matrix}] 路 [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ so, we will take $\overset{鈫�}{v} = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

By using the above formula we get:

role="math" localid="1659545669910"

\begin{array}{rcl} | x + 2 y + 3 z | & 猢� & \sqrt{14} \\ - \sqrt{14} & 猢� & x + 2 y + 3 z 猢� \sqrt{14} \\ x + 2 y + 3 z & 猢� & - \sqrt{14} \end{array}

Also note that equality holds only and only if $\overset{鈫�}{u}$ and $\overset{鈫�}{v}$ are linearly dependent.

So, $\overset{鈫�}{u} = 位 \overset{鈫�}{v}$ .

Thus,

$\overset{鈫�}{u} = [\begin{matrix} 位 \\ 2 位 \\ 3 位 \end{matrix}]$

Find the vector

Consider the equation:

$x^{2} + y^{2} + z^{2} = 位^{2} + 4 位^{2} + 9 位^{2} 鈬� 1 = 14 位^{2} 鈬� 位^{2} = \frac{1}{14} 鈬� 位 = 卤 \frac{1}{\sqrt{14}}$

Thus, the value is $位 = - \frac{1}{\sqrt{14}}$ as we have to obtain vector corresponding to minimal length.

Hence, the vector with the minimal length will be $\overset{鈫�}{u} = [\begin{matrix} - 1 / \sqrt{14} \\ - 2 / \sqrt{14} \\ - 3 / \sqrt{14} \end{matrix}]$

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91影视

Among all the unit vectors $\overset{鈫�}{u} = [\begin{matrix} x \\ y \\ z \end{matrix}]$ in $R^{3}$ , find the
one for which the sum role="math" localid="1659544842303" $x + 2 y + 3 z$ is minimal.

Short Answer

Step by step solution

Step by step solution Step 1: Use Cauchy-Schwarz inequality

Find the vector

One App. One Place for Learning.

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91影视

Among all the unit vectors u鈫�=[xyz]in R3, find theone for which the sum role="math" localid="1659544842303" x+2y+3z is minimal.

Short Answer

Step by step solution

Step by step solution Step 1: Use Cauchy-Schwarz inequality

Find the vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Applied Mathematics

Calculus

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Among all the unit vectors $\overset{鈫�}{u} = [\begin{matrix} x \\ y \\ z \end{matrix}]$ in $R^{3}$ , find the
one for which the sum role="math" localid="1659544842303" $x + 2 y + 3 z$ is minimal.