Q .12 - Calculus

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 $\mathbf{v}·\mathbf{i}=0$.

Step-by-Step Solution

Answer

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

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

$\mathbf{v}·\mathbf{i}=0$

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

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

$1 a=0$$The equation \left(1\right) is satisfied for a=0$

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

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