The equation $d=\frac{|Ax+By+Cz+D|}{(\sqrt{{A}^{2}+{B}^{2}+{C}^{2}})}$ gives is the distance between a plane and a point as the value 'd'. What is the unit of this value? Why is the distance between the plane and the point not a vector?

Kendrick Pierce
2022-05-22
Answered

The equation $d=\frac{|Ax+By+Cz+D|}{(\sqrt{{A}^{2}+{B}^{2}+{C}^{2}})}$ gives is the distance between a plane and a point as the value 'd'. What is the unit of this value? Why is the distance between the plane and the point not a vector?

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Haleigh Vega

Answered 2022-05-23
Author has **13** answers

The equation $d=\frac{|Ax+By+Cz+D|}{\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}}$ gives us the distance a point from a plane in ${\mathbb{R}}^{3}$. Here we discuss about Euclidean space and distances measured by physical units, here obey on $x$, $y$ and $z$ and $A$, $B$, $C$ and $D$ are constant scalars belong to ${\mathbb{R}}^{3}$ without units. Then the unit of d is the same as variables have.

Also, the distance is not a vector, it is the smallest way between the point and a point on plane measured without direction.

Also, the distance is not a vector, it is the smallest way between the point and a point on plane measured without direction.

Briana Petty

Answered 2022-05-24
Author has **3** answers

Distances have units of length. You can see this from the formula, where the numerator has units like $Ax$ and the denominator has units like $A$, so the ratio has units of $x$. You can certainly define a vector from the point to the nearest point on the plane, but that is not this formula. This formula gives the magnitude of that vector.

asked 2021-01-08

You buy 50 shares of stock for $18.25 per share. One month later, the value of the stock is $18.98 per share.

a. The value of the stock continues to increase by the same dollar amount each month. How much will your investment be worth in 1 year?

b. The value of the stock continues to increase by the same percent each month. How much will your investment be worth in 1 year?

a. The value of the stock continues to increase by the same dollar amount each month. How much will your investment be worth in 1 year?

b. The value of the stock continues to increase by the same percent each month. How much will your investment be worth in 1 year?

asked 2021-10-08

How many permutations of the letters ABCDEFG contain the string BCD?

asked 2022-07-26

Write the numeral 0.0284 x 10^4 in scientific notation

A. 284

B. 28.4 x 10 ^1

C. 2.84 x 10^2

D 0.284 x 10^3

A. 284

B. 28.4 x 10 ^1

C. 2.84 x 10^2

D 0.284 x 10^3

asked 2022-08-07

How prove this inequality $\sum _{cyc}\frac{{a}^{4}}{{a}^{2}+2{b}^{4}}\ge 1$

Let $a,b,c>0$ such $a+b+c=3$. Show that

$\frac{{a}^{4}}{{a}^{2}+2{b}^{4}}}+{\displaystyle \frac{{b}^{4}}{{b}^{2}+2{c}^{4}}}+{\displaystyle \frac{{c}^{4}}{{c}^{2}+2{a}^{4}}}\ge 1$

My attempt is to use Cauchy-Schwarz inequality. Hence, I consider

$({\displaystyle \frac{{a}^{4}}{{a}^{2}+2{b}^{4}}}+{\displaystyle \frac{{b}^{4}}{{b}^{2}+2{c}^{4}}}+{\displaystyle \frac{{c}^{4}}{{c}^{2}+2{a}^{4}}})({a}^{2}+{b}^{2}+{c}^{2}+2{a}^{4}+2{b}^{4}+2{c}^{4})\ge ({a}^{2}+{b}^{2}+{c}^{2}{)}^{2}$

However,

$({a}^{2}+{b}^{2}+{c}^{2}{)}^{2}\le ({a}^{2}+{b}^{2}+{c}^{2}+2{a}^{4}+2{b}^{4}+2{c}^{4})$

Let $a,b,c>0$ such $a+b+c=3$. Show that

$\frac{{a}^{4}}{{a}^{2}+2{b}^{4}}}+{\displaystyle \frac{{b}^{4}}{{b}^{2}+2{c}^{4}}}+{\displaystyle \frac{{c}^{4}}{{c}^{2}+2{a}^{4}}}\ge 1$

My attempt is to use Cauchy-Schwarz inequality. Hence, I consider

$({\displaystyle \frac{{a}^{4}}{{a}^{2}+2{b}^{4}}}+{\displaystyle \frac{{b}^{4}}{{b}^{2}+2{c}^{4}}}+{\displaystyle \frac{{c}^{4}}{{c}^{2}+2{a}^{4}}})({a}^{2}+{b}^{2}+{c}^{2}+2{a}^{4}+2{b}^{4}+2{c}^{4})\ge ({a}^{2}+{b}^{2}+{c}^{2}{)}^{2}$

However,

$({a}^{2}+{b}^{2}+{c}^{2}{)}^{2}\le ({a}^{2}+{b}^{2}+{c}^{2}+2{a}^{4}+2{b}^{4}+2{c}^{4})$

asked 2022-05-23

I was thinking of the example in Folland on page 61 i.e. $\mu (\mathbb{R})=\mathrm{\infty}$. Let ${f}_{n}=n{\chi}_{[0,1/n]}\to 0$ a.e. Then ${f}_{n}\to 0$ in measure. My inclination is that this is true so it requires proof. Also the same question is asked but for when $\mu (X)=1$. My inclination is that this is not true and a counterexample can be provided.

asked 2022-05-26

Is this a mistake on my part or theirs?

I'm not sure if I'm the one making the mistake, or my math book. It looks like the negative sign completely disappeared.

$\frac{3{x}^{2}}{-\sqrt{18}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{3{x}^{2}\sqrt{2}}{-\sqrt{36}}=\frac{3{x}^{2}\sqrt{2}}{6}=\frac{{x}^{2}\sqrt{2}}{2}$

Also, I am new to Stackexchange, so tell me if I am doing something wrong.

I'm not sure if I'm the one making the mistake, or my math book. It looks like the negative sign completely disappeared.

$\frac{3{x}^{2}}{-\sqrt{18}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{3{x}^{2}\sqrt{2}}{-\sqrt{36}}=\frac{3{x}^{2}\sqrt{2}}{6}=\frac{{x}^{2}\sqrt{2}}{2}$

Also, I am new to Stackexchange, so tell me if I am doing something wrong.

asked 2021-08-08

Find an equation for the set of all points equidistant from the point (0,0,2) and the xy-plane.