Chapter 10: Q .11 (page 811)

What geometric relationship must two vectors have in order for $‖ u + v ‖ = ‖ u ‖ + ‖ v ‖$ ?

Short Answer

Expert verified

The geometric relationship between the vectors is that they are parallel to each other and are in same direction.

Step by step solution

Step 1:Given information

The given expression is $‖ u + v ‖ = ‖ u ‖ + ‖ v ‖$

Step 2:Simplification

$Consider the non-zero vectors u and v .$

$The objective is to find the geometric relationship between two vectors if ‖ u + v ‖ = ‖ u ‖ + ‖ v ‖ .$

$To find the relationship, consider the value ‖ u + v ‖ .$

$The expression ‖ u + v ‖ gives:$

$‖ u + v ‖^{2} = (u + v) \cdot (u + v)$

$‖ u + v ‖^{2} = ‖ u ‖^{2} + ‖ v ‖^{2} + 2 ‖ u ‖ ‖ v ‖ \cos θ \dots \dots$

$For θ = 0 °, the equation (1) reduces to:$

$‖ u + v ‖^{2} = ‖ u ‖^{2} + ‖ v ‖^{2} + 2 ‖ u ‖ v ‖$

$‖ u + v ‖^{2} = (‖ u ‖ + ‖ v ‖)^{2}$

$‖ u + v ‖ = ‖ u ‖ + ‖ v ‖$

$The result ‖ u + v ‖ = ‖ u ‖ + ‖ v ‖ holds when θ = 0 ° .$

Therefore, the geometric relationship between the vectors is that they are parallel to each other and are in same direction.

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What geometric relationship must two vectors have in order for $‖ u + v ‖ = ‖ u ‖ + ‖ v ‖$ ?

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

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What geometric relationship must two vectors have in order for‖u+v‖=‖u‖+‖v‖ ?

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Geometry

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Study anywhere. Anytime. Across all devices.

What geometric relationship must two vectors have in order for $‖ u + v ‖ = ‖ u ‖ + ‖ v ‖$ ?