Question

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

Short answer

$\text{The position vector should lie on}yz\text{-plane so that}\mathbf{v}·\mathbf{i}=0\text{holds.}$

Step-by-step solution

Step 1:Given information

$\mathbf{i}=⟨1,0,0⟩$in${\mathrm{ℝ}}^3$

Step 2:Explaination

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

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

$\text{The dot product of}\mathbf{v}=⟨a,b,c⟩\text{and}\mathbf{i}=⟨1,0,0⟩\text{is}0\text{.}$

Therefore

$\mathbf{v}·\mathbf{i}=0$

$⟨a,b,c⟩·⟨1,0,0⟩=0\text{(Substitution)}$

$1a+0b+0c=0\text{(Simplify)}$

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

$\text{The equation (1) is satisfied for}a=0\text{.}$

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

$\text{Thus, the position vector should lie on}yz\text{-plane so that}\mathbf{v}·\mathbf{i}=0\text{holds.}$