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Calculus / Calculus: Early Transcendentals 1 / Chapter 11.3 / Problem 47E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Answer: Vectors with equal projections Given a fixed

Chapter 11, Problem 47E

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QUESTION:

Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with the same value of \(\mathrm{proj}_\mathbf{v} \mathbf{u}\).

Let \(\mathbf{v}=\langle 0,0,1\rangle\). Give a description of all position vectors u such that \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}=\operatorname{proj}_{\mathbf{v}}\langle 1,2,3\rangle\).

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QUESTION:

Vectors with equal projections Given a fixed vector v, there is an infinite set of vectors u with the same value of \(\mathrm{proj}_\mathbf{v} \mathbf{u}\).

Let \(\mathbf{v}=\langle 0,0,1\rangle\). Give a description of all position vectors u such that \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}=\operatorname{proj}_{\mathbf{v}}\langle 1,2,3\rangle\).

ANSWER:

Solution 47E

Step 1:

In this problem we need to find the description of all position vectors $u$ such that $pro{j}_{v}u=pro{j}_{v}〈1,2,3〉$ where $v\ =\ 〈0,0,1〉$

Let $u=\ 〈a,b,c〉$

Projection of u onto v is $pro{j}_{v}\ u=\frac{u.v}{|v{|}^{2}}\ v$

Projection of $〈1,2,3〉$ onto v is $pro{j}_{v}\ u=\frac{〈1,2,3〉.v}{|v{|}^{2}}\ v$

We have $pro{j}_{v}u=pro{j}_{v}〈1,2,3〉$

$\Rightarrow{}\frac{u.v}{|v{|}^{2}}\ v\ =\frac{〈1,2,3〉.v}{|v{|}^{2}}\ v$

$\Rightarrow{}u.v=〈1,2,3〉.v$

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