Chapter 10: Q .13. (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 $v \cdot i = 0$ .

Short Answer

Expert verified

$The position vector should lie on y z -plane so that v \cdot i = 0 holds.$

Step by step solution

Step 1:Given information

$i = ⟨ 1, 0, 0 ⟩$ in $ℝ^{3}$

Step 2:Explaination

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

$Assume that the vector vin ℝ^{3} has the position vector v = ⟨ a, b, c ⟩ .$

$The dot product of v = ⟨ a, b, c ⟩ and i = ⟨ 1, 0, 0 ⟩ is 0 .$

Therefore

$v \cdot i = 0$

$⟨ a, b, c ⟩ \cdot ⟨ 1, 0, 0 ⟩ = 0 (Substitution)$

$1 a + 0 b + 0 c = 0 (Simplify)$

$1 a = 0$ ....equaton 1

$The equation (1) is satisfied for a = 0 .$

$The vector v in ℝ^{3} has the position vector v = ⟨ 0, b, c ⟩ so that v \cdot i = 0 holds.$

$Thus, the position vector should lie on y z -plane so that v \cdot i = 0 holds.$

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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 $v \cdot i = 0$ .

Short Answer

Step by step solution

Step 1:Given information

Step 2:Explaination

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Consider the position vectori=⟨1,0,0⟩in ℝ3Describe the set of position vectors vin $ℝ3with the property thatv·i=0.

Short Answer

Step by step solution

Step 1:Given information

Step 2:Explaination

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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 $v \cdot i = 0$ .