Question

12. Consider the position vector $\mathbf{i}=⟨1,0⟩$in${\mathrm{ℝ}}^2$. Describe the set of position vectors $\mathit{v}$in ${\mathrm{ℝ}}^2$with the property that $\mathbf{v}·\mathbf{i}=0$.

Short answer

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

Step-by-step solution

Step 1:Given information

$\mathbf{i}=⟨1,0⟩$in${\mathrm{ℝ}}^2$

Step 2:Explaination

$\text{Consider the position vector}\mathbf{i}=⟨1,0⟩\text{in}{\mathrm{ℝ}}^2\text{.}$

$\text{Assume that the vector}\mathbf{v}\text{in}{\mathrm{ℝ}}^2\text{has the position vector}\mathbf{v}=⟨a,b⟩$

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

Therefore

$\mathbf{v}·\mathbf{i}=0$

$⟨a,b⟩·⟨1,0⟩=0\text{(Substitution)}$

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

$1a=0$$Theequation\left(1\right)issatisfiedfora=0$

$\text{The vector}\mathbf{v}\text{in}{\mathrm{ℝ}}^2\text{has the position vector}\mathbf{v}=⟨0,b⟩\text{so that}\mathbf{v}·\mathbf{i}=0\text{holds.}$

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