Mathematics, 13.02.2020 21:38 leahstubbs

Parameterize the plane that contains the three points (−3,1,−2), (−6,−10,−4) and (15,5,20).
(Use s and t for the parameters in your parameterization)

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The plane that contains the points (-3, 1, -2), (-6, -10, -4) and (15, 5, 20) is described by $\vec r = (-3,1,-2) + s\cdot (-3,-11,-2)+t\cdot (18, 4,22)$ .

Let be A(x, y,z) = (-3, 1, -2) , B(x,y,z) = (-6, -10, -4) and C(x,y,z) = (15, 5, 20) , the equation of the plane in parametric form is:

$\vec r = A(x,y,z) + s\cdot \overrightarrow{AB} + t \cdot \overrightarrow{AC}$ (1)

Where and are parameters and:

$\overrightarrow{AB} = B(x,y,z) - A(x,y,z)$ (2)

$\overrightarrow{AC} = C(x,y,z)-A(x,y,z)$ (3)

If we know that A(x, y,z) = (-3, 1, -2) , B(x,y,z) = (-6, -10, -4) and C(x,y,z) = (15, 5, 20) , then the parametric form is:

$\overrightarrow{AB} = (-6,-10,-4)-(-3,1,-2)$

$\overrightarrow{AB} = (-3,-11, -2)$

$\overrightarrow{AC} = (15,5,20)-(-3,1,-2)$

$\overrightarrow{AC} = (18, 4, 22)$

$\vec r = (-3,1,-2) + s\cdot (-3,-11,-2)+t\cdot (18, 4,22)$

The plane that contains the points (-3, 1, -2), (-6, -10, -4) and (15, 5, 20) is described by $\vec r = (-3,1,-2) + s\cdot (-3,-11,-2)+t\cdot (18, 4,22)$ .

We kindly invite to check this question on parametric formulas: link

Answer from: Yangster9305

First find any two vectors in the plane, and take their cross product. The cross product will be perpendicular to both of them.

To find any vectors in the plane, take any two of the given points at a time, and treating them as vectors, subtract them. These vector differences are parallel to the plane we want to find.

(-3, 1, -2) - (-6, -10, -4) = (3, 11, 2)

(-3, 1, -2) - (15, 5, 20) = (-18, -4, -22)

Take the cross product to get the normal vector to the plane:

(3, 11, 2) x (-18, -4, -22) = (-234, 30, 186)

Let (x, y, z) be any point on the plane. The vector

(x, y, z) - (-3, 1, -2) = (x + 3, y - 1, z + 2)

runs parallel to the plane, so it's perpendicular to the plane's normal, which means their dot product is 0. This gives us the Cartesian equation for the plane,

(-234, 30, 186) • (x + 3, y - 1, z + 2) = 0

-234(x + 3) + 30(y - 1) + 186(z + 2) = 0

-234x + 30y + 186 = 360

Every coefficient has a GCD of 6, so the equation is equivalent to

-39x + 5y + 31z = 60

Finally - and this is the easiest step - write this in parametric form. Let x = s and y = t, then

z = (60 + 39s - 5t)/31

So the plane can be parameterized by

P(s, t) = (s, t, (60 + 39s - 5t)/31)

where s and t are any real numbers.

Answer from: Quest

answer: y = -12x + 108

step-by-step explanation: do (y = m x+b) formula.

Answer from: Quest

answer: 1,380

step-by-step explanation:

1: you turn %15 into a decimal by moving the decimal point two places to the left.

2: and then you multiply .15 by 1,200 , which will get you 180.00

3: from there you will add 180 to 1,200

4: which will get you 1,380

hope it *im kind of slow so idk if my answer is right*

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Parameterize the plane that contains the three points (−3,1,−2), (−6,−10,−4) and (15,5,20).
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