Question

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${proj}_{\mathrm{i}}\mathbf{v}=\mathbf{i}$.

Short answer

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

Step-by-step solution

Step 1:Given information

the position vector in${\mathrm{ℝ}}^2$

Step 2:Simplification

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

$\text{The vector projection of}\mathbf{v}\text{onto u is given by:}$

${proj}_{\mathbf{u}}\mathbf{v}=\frac{\mathbf{u}·\mathbf{v}}{\Vert \mathbf{u}{\Vert }^2}\mathbf{u}$

$\text{Assume the position vectors of}\mathbf{v}=⟨a,b⟩\text{and}{proj}_i\mathbf{v}=\mathbf{i}\text{.}$

Therefore

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

$\frac{\mathbf{1}·\mathbf{v}}{\Vert \mathbf{i}{\Vert }^2}\mathbf{i}=\mathbf{i}\text{(Substitution)}$

$\frac{⟨1,0⟩·⟨a,b⟩}{\left(1^2+0^2\right)}\mathbf{i}=\mathbf{i}(\text{Simplify)}$

$1a=1$

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

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

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