71 Under what conditions does the ratio A}/B equal A_x//B_x

[karush]

71.15 Two vectors $\vec{A}$ and $\vec{B}$ lie in xy plane.
Under what conditions does the ratio $\vec{A}/\vec{B}$ equal $\vec{A_x}/\vec{B_x}$?

Sorry but I had a hard time envisioning what this would be?
also thot I posted this earlier but I can't find it

 

[MarkFL]

I'm going to assume we're talking about the ratio of magnitudes. Suppose:

\(\displaystyle \vec{A}=\left\langle A_x,A_y \right\rangle\)

\(\displaystyle \vec{B}=\left\langle B_x,B_y \right\rangle\)

Then, let's see what happens when we write:

\(\displaystyle \frac{A_x^2+A_y^2}{B_x^2+B_y^2}=\frac{A_x^2}{B_x^2}\)

\(\displaystyle A_x^2B_x^2+A_y^2B_x^2=A_x^2B_x^2+A_x^2B_y^2\)

\(\displaystyle A_y^2B_x^2=A_x^2B_y^2\)

\(\displaystyle \frac{A_y^2}{A_x^2}=\frac{B_y^2}{B_x^2}\)

\(\displaystyle \frac{A_y}{A_x}=\pm\frac{B_y}{B_x}\)

What conclusion may we draw from this result?

 

[HallsofIvy]

I would immediately have a problem with $$\frac{\vec{A}}{\vec{B}}$$. The division of vectors is not defined. Did you mean $$\frac{|\vec{A}|}{|\vec{B}|}$$? That would be equal to $$\frac{A_x}{B_x}$$ if and only if the other components of $$\vec{A}$$ and $$\vec{B}$$ are 0.