Q .14. - Calculus

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

Step-by-Step Solution

Answer

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

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 

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

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

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

$1a=1$

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

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

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