PLZZZ HELP ILL GIVE BRAINLIEST

The equation giving a family of ellipsoids is u = (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) . Find the unit vector normal to each

Fynjy0 [20]

Answer:

$\hat{n}\ =\ \ \dfrac{\dfrac{x}{a^2}\hat{i}+\ \dfrac{y}{b^2}\hat{j}+\ \dfrac{z}{c^2}\hat{k}}{\sqrt{(\dfrac{x}{a^2})^2+(\dfrac{y}{b^2})^2+(\dfrac{z}{c^2})^2}}$

Step-by-step explanation:

Given equation of ellipsoids,

$u\ =\ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}$

The vector normal to the given equation of ellipsoid will be given by

$\vec{n}\ =\textrm{gradient of u}$

$=\bigtriangledown u$

$=\ (\dfrac{\partial{}}{\partial{x}}\hat{i}+ \dfrac{\partial{}}{\partial{y}}\hat{j}+ \dfrac{\partial{}}{\partial{z}}\hat{k})(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2})$

$=\ \dfrac{\partial{(\dfrac{x^2}{a^2})}}{\partial{x}}\hat{i}+\dfrac{\partial{(\dfrac{y^2}{b^2})}}{\partial{y}}\hat{j}+\dfrac{\partial{(\dfrac{z^2}{c^2})}}{\partial{z}}\hat{k}$

$=\ \dfrac{2x}{a^2}\hat{i}+\ \dfrac{2y}{b^2}\hat{j}+\ \dfrac{2z}{c^2}\hat{k}$

Hence, the unit normal vector can be given by,

$\hat{n}\ =\ \dfrac{\vec{n}}{\left|\vec{n}\right|}$

$=\ \dfrac{\dfrac{2x}{a^2}\hat{i}+\ \dfrac{2y}{b^2}\hat{j}+\ \dfrac{2z}{c^2}\hat{k}}{\sqrt{(\dfrac{2x}{a^2})^2+(\dfrac{2y}{b^2})^2+(\dfrac{2z}{c^2})^2}}$

$=\ \dfrac{\dfrac{x}{a^2}\hat{i}+\ \dfrac{y}{b^2}\hat{j}+\ \dfrac{z}{c^2}\hat{k}}{\sqrt{(\dfrac{x}{a^2})^2+(\dfrac{y}{b^2})^2+(\dfrac{z}{c^2})^2}}$

Hence, the unit vector normal to each point of the given ellipsoid surface is

$\hat{n}\ =\ \ \dfrac{\dfrac{x}{a^2}\hat{i}+\ \dfrac{y}{b^2}\hat{j}+\ \dfrac{z}{c^2}\hat{k}}{\sqrt{(\dfrac{x}{a^2})^2+(\dfrac{y}{b^2})^2+(\dfrac{z}{c^2})^2}}$

3 0

3 years ago

If a number is doubled then which of

Tamiku [17]

Answer: Its cube is six times the

cube of the given number.

Step-by-step explanation:

Let's assume we have a number which is 2. The cube will be:

= 2³

= 2 × 2 × 2

= 8

Let's now assume, we double the number 2, we'll have 2 × 2 = 4. Therefore, the cube will be:

= 4³

= 4 × 4 × 4

= 64

Therefore, from the explanation above, 2 raise to power 6 is 64. This means, if a number is doubled, then its cube is six times the cube of the given number.

6 0

3 years ago

Phil has 3 times as many rocks as Phil. They have 48 rocks total. How many rocks does phil have than Peter ?

PilotLPTM [1.2K]

U have to multiply 48*3

4 0

3 years ago

You receive an 11 percent commission on all sales you make if your commission for the month is 512.50 dollars what was your tota

JulijaS [17]

Total sales were $4659.1

Step-by-step explanation:

Given,

Commission = 11%

Amount of commission = $512.50

Let,

total sales = x

Therefore,

11% of x = 512.50

$\frac{11}{100}x=512.50\\0.11x=512.50$

Dividing both sides by 0.11

$\frac{0.11x}{0.11}=\frac{512.50}{0.11}\\x= $4659.1$

Total sales were $4659.1

Keywords: percentage, division

Learn more about division at:

#LearnwithBrainly

4 0

3 years ago

Fallon collected data for her favorite TV show. She recorded the number of viewers and the number of weeks since the show premie

adelina 88 [10]

Answer:

On the best-fit line for the points {(2,3) (5,7) (1,2) (4,8)}, one point is (0.5,1.25) and another is the y-intercept (0,0.5). Use the formula for the slope of a line, m = (y2 - y1)/(x2 - x1), to find the slope

Step-by-step explanation:

Hope it help

Mark me as brainliest

7 0

3 years ago

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