Q .13. - Calculus

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 $\mathbf{v}·\mathbf{i}=0$.

Step-by-Step Solution

Answer

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

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

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

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

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