Chapter 9: Q. 40 (page 613)

Let vand wdenote two nonzero vectors. Show that the vector $v - α w$ is orthogonal to wif $α = \frac{v \cdot w}{{||w||}^{2}}$

Short Answer

Expert verified

The given vectors are orthogonal.

Step by step solution

Step 1. Given Information

There are two nonzero vectorsvandw.

We have to show that the vectors $v - α w$ is orthogonal towif $α = \frac{v \cdot w}{{||w||}^{2}} .$

Step 2. Showing the vectors are orthogonal

As we know if any two vectors are orthogonal then their dot product is equal to zero.

Let's find the dot product of $(v - α w) and (w)$

So,

$(v - α w) \cdot (w) = v \cdot w - α {||w||}^{2} [a \cdot a = {||a||}^{2}] = v \cdot w - (\frac{v \cdot w}{{||w||}^{2}}) {||w||}^{2} = v \cdot w - v \cdot w = 0$

Therefore, given vectors are orthogonal.

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.

Let vand wdenote two nonzero vectors. Show that the vector $v - α w$ is orthogonal to wif $α = \frac{v \cdot w}{{||w||}^{2}}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Showing the vectors are orthogonal

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Probability and Statistics

Calculus

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Let vand wdenote two nonzero vectors. Show that the vectorv-αwis orthogonal to wifα=v·ww2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Showing the vectors are orthogonal

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Probability and Statistics

Calculus

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Let vand wdenote two nonzero vectors. Show that the vector $v - α w$ is orthogonal to wif $α = \frac{v \cdot w}{{||w||}^{2}}$