Chapter 10: Q .15. (page 811)

Consider the position vector $i = ⟨ 1, 0, 0 ⟩$ in $ℝ^{3}$ . Describe the set of position vectors $v$ in $ℝ^{3}$ with the property that proi, $v = i$ .

Short Answer

Expert verified

$Any position vector of the type v = ⟨ 1, b, c ⟩ {satisfies the conditionproj}_{i} v = i .$

Step by step solution

Step 1:Given information

The position vector $i = ⟨ 1, 0, 0 ⟩$ in $ℝ^{3}$

Step 2:Simplification

$Consider the position vector i = ⟨ 1, 0, 0 ⟩ in ℝ^{3} .$

$The vector projection of v onto u is given by:$

${proj}_{u} v = \frac{u \cdot v}{‖ u ‖^{2}} u$

$Assume the position vectors of v = ⟨ a, b, c ⟩ and {proj}_{i} v = i .$

Therefore

${proj}_{i} v = i (Given condition)$

$\frac{i \cdot v}{‖ i ‖^{2}} i = i (Substitution)$

$\frac{(1, 0, 0 ⟩ \cdot ⟨ a, b, c ⟩}{(1^{2} + 0^{2} + 0^{2})} i = i (Simplify)$

$1 a = 1$ equation 1

$The equation (1) is satisfied for a = 1 .$

$The vector v in ℝ^{3} has the position vector v = ⟨ 1, b, c ⟩ so that proji v = i holds.$

$Thus, any position vector of the type v = ⟨ 1, b, c ⟩ {satisfies the conditionproj}_{i} v = i .$

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Consider the position vector $i = ⟨ 1, 0, 0 ⟩$ in $ℝ^{3}$ . Describe the set of position vectors $v$ in $ℝ^{3}$ with the property that proi, $v = i$ .

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

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Consider the position vectori=⟨1,0,0⟩in ℝ3. Describe the set of position vectors vin ℝ3with the property that proi,v=i.

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

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Consider the position vector $i = ⟨ 1, 0, 0 ⟩$ in $ℝ^{3}$ . Describe the set of position vectors $v$ in $ℝ^{3}$ with the property that proi, $v = i$ .