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

  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Conditions Ratio
AI Thread Summary
The discussion centers on the conditions under which the ratio of two vectors, A/B, equals the ratio of their x-components, A_x/B_x. It is established that if A and B are defined in the xy-plane, the equality holds when the y-components of both vectors are zero. The mathematical derivation shows that the ratios of the y-components to the x-components must be equal, leading to the conclusion that A_y/A_x equals B_y/B_x. The conversation highlights the confusion around vector division, clarifying that the intended interpretation involves the magnitudes of the vectors rather than direct vector division. Thus, the condition for the ratios to be equal is that the y-components must be zero.
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
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
 
Mathematics news on Phys.org
I'm going to assume we're talking about the ratio of magnitudes. Suppose:

$$\vec{A}=\left\langle A_x,A_y \right\rangle$$

$$\vec{B}=\left\langle B_x,B_y \right\rangle$$

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

$$\frac{A_x^2+A_y^2}{B_x^2+B_y^2}=\frac{A_x^2}{B_x^2}$$

$$A_x^2B_x^2+A_y^2B_x^2=A_x^2B_x^2+A_x^2B_y^2$$

$$A_y^2B_x^2=A_x^2B_y^2$$

$$\frac{A_y^2}{A_x^2}=\frac{B_y^2}{B_x^2}$$

$$\frac{A_y}{A_x}=\pm\frac{B_y}{B_x}$$

What conclusion may we draw from this result?
 
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.
 
Last edited by a moderator:
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top