Problem 65 Hard Difficulty

If $ \theta $ is the angle between vectors $ a $ and $ b $, show that
$$ proj_a b \cdot proj_b a = (a \cdot b) \cos^2 \theta $$

Answer

$\operatorname{Proj}_{\mathbf{a}} \mathbf{b} \cdot \operatorname{Proj}_{\mathbf{b}} \mathbf{a}=\frac{|\mathbf{b}| \cos \theta}{|\mathbf{a}|} \mathbf{a} \cdot \frac{|\mathbf{a}| \cos \theta}{|\mathbf{b}|} \mathbf{b}=\mathbf{a} \cdot \mathbf{b} \cos ^{2} \theta$

Topics

Vectors

Calculus: Early Transcendentals

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Discussion

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Video Transcript

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Wen Z.

Topics

Vectors

Calculus: Early Transcendentals

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Problem 65 Hard Difficulty

If $ \theta $ is the angle between vectors $ a $ and $ b $, show that
$$ proj_a b \cdot proj_b a = (a \cdot b) \cos^2 \theta $$

Answer

$\operatorname{Proj}_{\mathbf{a}} \mathbf{b} \cdot \operatorname{Proj}_{\mathbf{b}} \mathbf{a}=\frac{|\mathbf{b}| \cos \theta}{|\mathbf{a}|} \mathbf{a} \cdot \frac{|\mathbf{a}| \cos \theta}{|\mathbf{b}|} \mathbf{b}=\mathbf{a} \cdot \mathbf{b} \cos ^{2} \theta$

Topics

Calculus: Early Transcendentals

Discussion

Top Calculus 3 Educators

Caleb E.

Samuel H.

Michael J.

Joseph L.

Calculus 3 Bootcamp

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 12

Video Transcript

Topics

Calculus: Early Transcendentals

Top Calculus 3 Educators

Caleb E.

Samuel H.

Michael J.

Joseph L.

Calculus 3 Bootcamp

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Problem 65 Hard Difficulty

If $ \theta $ is the angle between vectors $ a $ and $ b $, show that$$ proj_a b \cdot proj_b a = (a \cdot b) \cos^2 \theta $$

Answer

$\operatorname{Proj}_{\mathbf{a}} \mathbf{b} \cdot \operatorname{Proj}_{\mathbf{b}} \mathbf{a}=\frac{|\mathbf{b}| \cos \theta}{|\mathbf{a}|} \mathbf{a} \cdot \frac{|\mathbf{a}| \cos \theta}{|\mathbf{b}|} \mathbf{b}=\mathbf{a} \cdot \mathbf{b} \cos ^{2} \theta$

Topics

Calculus: Early Transcendentals

Discussion

Top Calculus 3 Educators

Caleb E.

Samuel H.

Michael J.

Joseph L.

Calculus 3 Bootcamp

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 12

Video Transcript

Topics

Calculus: Early Transcendentals

Top Calculus 3 Educators

Caleb E.

Samuel H.

Michael J.

Joseph L.

Calculus 3 Bootcamp

Vectors Intro

Vector Basics - Example 1

Recommended Videos

If $ \theta $ is the angle between vectors $ a $ and $ b $, show that
$$ proj_a b \cdot proj_b a = (a \cdot b) \cos^2 \theta $$