Chapter 11: Q. 1 (page 897)

Projecting one vector onto another: Show that the formula for the projection of a vector v onto a nonzero vector u is given by $p r o j_{u} v = \frac{u . v}{| |u| |}, w h e r e u \neq 0 .$

Short Answer

Expert verified

$D o t p r o d u c t : u . v = |u| |v| \cos θ, p r o j_{u} v = |v| \cos θ, S o l v i n g a b o v e t w o e q u a t i o n s w e g e t, p r o j_{u} v = \frac{u . v}{||u||} .$

Step by step solution

Step 1. Given Information.

Given two vectors uand vwhere u is a nonzero vector.

Step 2. Dot product.

$u . v = |u| . |v| \cos θ$

Step 3. Proof

$L e t s u p p o s e w e h a v e a v e c t o r v a n d a n o t h e r v e c t o r u a n d t h e s e v e c t o r s m a k e a n a n g l e o f θ b e t w e e n t h e m, N o w, t h e w e t a k e t h e c o m p o n e n t o f t h e v a l o n g t h e u w e g e t c o m p_{u} v = |v| \cos θ, N o w t h e p r o j e c t i o n i s t h e s a m e a s t h e c o m p o n e n t w e c a n w r i t e, p r o j_{u} v = c o m p_{u} v = |v| \cos θ N o w m u l t i p l y d i v i d e b y t h e m a g n i t u d e o f u v e c t o r i n t h e n u m e r a t o r a n d d e n o m i n a t o r i n t h e R H S, w e g e t, p r o j_{u} v = \frac{|v| |u| \cos θ}{|u|}, N o w f r o m d o t p r o d u c t w e g e t, p r o j_{u} v = \frac{u . v}{||u||} .$

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.

Projecting one vector onto another: Show that the formula for the projection of a vector v onto a nonzero vector u is given by $p r o j_{u} v = \frac{u . v}{| |u| |}, w h e r e u \neq 0 .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Dot product.

Step 3. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Discrete Mathematics

Statistics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Projecting one vector onto another: Show that the formula for the projection of a vector v onto a nonzero vector u is given byprojuv=u.v|u|,whereu≠0.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Dot product.

Step 3. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Discrete Mathematics

Statistics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Projecting one vector onto another: Show that the formula for the projection of a vector v onto a nonzero vector u is given by $p r o j_{u} v = \frac{u . v}{| |u| |}, w h e r e u \neq 0 .$