Chapter 9: Q. 21 (page 612)

In Problems 19–24, decompose v into two vectors $v_{1}$ and $v_{2}$ , where $v_{1}$ is parallel to w and $v_{2}$ is orthogonal to w.
$v = i - j w = - i - 2 j$

Short Answer

Expert verified

$v_{1} = - 0.2 i - 0.4 j v_{2} = 1.2 i - 0.6 j$

Step by step solution

Step 1. Concept Used

The decomposition of $v$ into $v_{1}$ and $v_{2}$ , where $v_{1}$ is parallel to $w$ and localid="1646827028342" $v_{2}$ is orthogonal to $w$ , is

localid="1646827499665" $v_{1} = \frac{v \cdot w}{{|w|}^{2}} w v_{2} = v - v_{1}$

Step 2. Find the dot product and w

For the vectors $v = i - j$ and $w = - i - 2 j$ the dot product is given as

$v \cdot w = (i - j) \cdot (- i - 2 j) v \cdot w = 1 \cdot (- 1) + (- 1) \cdot (- 2) v \cdot w = - 1 + 2 v \cdot w = 1$

The magnitude of vector $w = - i - 2 j$ is given as

$|w| = \sqrt{{(- 1)}^{2} + {(- 2)}^{2}} |w| = \sqrt{1 + 4} |w| = \sqrt{5}$

Step 3. Find v1 and v2

The vector $v_{1}$ which is parallel to $w$ is given by the formula $v_{1} = \frac{v \cdot w}{{|w|}^{2}} w$ .

On substituting the calculated values we get

$v_{1} = \frac{1}{{(\sqrt{5})}^{2}} (- i - 2 j) v_{1} = \frac{1}{5} (- i - 2 j) v_{1} = 0.2 (- i - 2 j) v_{1} = - 0.2 i - 0.4 j$

and the vector $v_{2}$ orthogonal to $w$ is found as

$v_{2} = v - v_{1} v_{2} = i - j - (- 0.2 i - 0.4 j) v_{2} = i - j + 0.2 i + 0.4 j v_{2} = (1 + 0.2) i + (- 1 + 0.4) j v_{2} = 1.2 i - 0.6 j$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 19–24, decompose v into two vectors $v_{1}$ and $v_{2}$ , where $v_{1}$ is parallel to w and $v_{2}$ is orthogonal to w.
$v = i - j w = - i - 2 j$

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Find the dot product and w

Step 3. Find v1 and v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Statistics

Applied Mathematics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 19–24, decompose v into two vectors v1and v2, where v1 is parallel to w and v2 is orthogonal to w. v=i-jw=-i-2j

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Find the dot product and w

Step 3. Find v1 and v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Statistics

Applied Mathematics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 19–24, decompose v into two vectors $v_{1}$ and $v_{2}$ , where $v_{1}$ is parallel to w and $v_{2}$ is orthogonal to w.
$v = i - j w = - i - 2 j$