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

• MHB
• karush
In summary, when considering the ratio of magnitudes of two vectors in the xy plane, the condition for it to equal the ratio of their x-components is that the y-components of both vectors must be 0. This is because the division of vectors is not defined and can only be considered when working with the magnitudes of the vectors.
karush
Gold Member
MHB
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

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?

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.

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## 1. What is the meaning of the ratio A/B?

The ratio A/B represents the relationship between two quantities, A and B. It is the result of dividing the value of A by the value of B. It can also be thought of as the number of times A is contained in B.

## 2. How is the ratio A/B calculated?

The ratio A/B is calculated by dividing the value of A by the value of B. For example, if A = 10 and B = 5, then the ratio A/B would be 10/5 = 2. This means that A is twice as large as B.

## 3. What does the ratio A_x/B_x represent?

The ratio A_x/B_x represents the relationship between two specific values of A and B, denoted by the subscripts x. This allows for the calculation of different ratios for different values of A and B.

## 4. Under what conditions does the ratio A/B equal A_x/B_x?

The ratio A/B will equal A_x/B_x when the values of A and B are the same as the values of A_x and B_x. In other words, when comparing the same quantities, the ratio will be the same regardless of the specific values used.

## 5. What are some real-life examples of using the ratio A/B?

The ratio A/B is commonly used in various fields such as finance, engineering, and science. Some examples include calculating financial ratios such as debt-to-equity ratio, using gear ratios in mechanics, and determining concentration ratios in chemistry.

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