Chapter 9: Q. 41 (page 613)

Let vand wdenote two nonzero vectors. Show that the vectors $||w|| v + ||v|| w a n d ||w|| v - ||v|| w$ are orthogonal.

Short Answer

Expert verified

The given vectors are orthogonal.

Step by step solution

Step 1. Given Information

There are two nonzero vectors vand w.

We have to show that the vectors $||w|| v + ||v|| w a n d ||w|| v - ||v|| w$ are orthogonal.

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 $||w|| v + ||v|| w a n d ||w|| v - ||v|| w .$

So,

role="math" localid="1646821115195" $(| | w | | v + | | v | | w) \cdot (| | w | | v - | | v | | w) = {| | w | |}^{2} v \cdot v - | | w | | | | v | | v \cdot w + | | v | | | | w | | w \cdot v - | | v | |^{2} w \cdot w = {| | w | |}^{2} v \cdot v - | | v | |^{2} w \cdot w = {| | w | |}^{2} | | v | |^{2} - | | v | |^{2} {| | w | |}^{2} [a \cdot a = {||a||}^{2}] = 0$

Therefore the given vectors are orthogonal.

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Let vand wdenote two nonzero vectors. Show that the vectors $||w|| v + ||v|| w a n d ||w|| v - ||v|| w$ are orthogonal.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Showing the vectors are orthogonal

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Let vand wdenote two nonzero vectors. Show that the vectorswv+vwandwv-vware orthogonal.

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

Probability and Statistics

Pure Maths

Calculus

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let vand wdenote two nonzero vectors. Show that the vectors $||w|| v + ||v|| w a n d ||w|| v - ||v|| w$ are orthogonal.