Find an equation of the plane.

The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)

The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)

Wotzdorfg
2021-06-03
Answered

Find an equation of the plane.

The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)

The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)

You can still ask an expert for help

au4gsf

Answered 2021-06-04
Author has **95** answers

To find the equation of the plane we need to find its normal vector.

Let us call the given points as: P(2,1,2), Q(3,-8,6), R(-2,-3,1)

Note that$\overrightarrow{PQ}$ and $\overrightarrow{PR}$ are vectors in the plane

Hence, their cross product will give us normal vector to the plane

$\overrightarrow{PQ}=(3,-8,6)-(2,1,2)=(3-2,-8-1,6-2)=(1,-9,4)$

$\overrightarrow{PR}=(-2,-3,1)-(2,1,2)=(-2-2,-3-1,1-2)=(-4,-4,-1)$

$n=\overrightarrow{PQ}\times \overrightarrow{PR}=\left|\begin{array}{ccc}i& j& k\\ 1& -9& 4\\ -4& -4& -1\end{array}\right|=(25,-15,-40)$

Equation of a plane passing through the point (a,b,c) and having normal vector (l,m,n) is

$l(x-a)+m(y-b)+n(z-c)=0$

We found the normal vector in the previous cell. For a point on the plane, we can choose any of the three given points, I will choose P(2,1,2)

Equation of the plane is

$25(x-2)-15\cdot (y-1)-40\cdot (z-2)]=0$

Divide throughout by 5

$5(x-2)-3\cdot (y-1)-8\cdot (z-2)=0$

$5x-3y-8z+9=0$

Result:

$5x-3y-8z+9=0$

Let us call the given points as: P(2,1,2), Q(3,-8,6), R(-2,-3,1)

Note that

Hence, their cross product will give us normal vector to the plane

Equation of a plane passing through the point (a,b,c) and having normal vector (l,m,n) is

We found the normal vector in the previous cell. For a point on the plane, we can choose any of the three given points, I will choose P(2,1,2)

Equation of the plane is

Divide throughout by 5

Result:

asked 2021-05-14

Use the given graph off over the interval (0, 6) to find the following.

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)

b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)

c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)

d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)

e) The coordinates of the point of inflection.$(x,\text{}y)=$

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)

b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)

c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)

d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)

e) The coordinates of the point of inflection.

asked 2022-02-02

How do you find the polar coordinates of the point $(-2,\text{}0)?$

asked 2021-10-16

Find a vector equation and parametric equations for the line segment that joins P to Q . P(2, 0, 0), Q(6, 2, -2)

asked 2021-11-30

For problems 1 the area of the region below the parameric curve given by the set of parametric equations. For each problem you may assume that each curve traces out exactly once from right to left for the given range of t. For these problems you should only use the given parametric equations to determine the answer.

1)$x={t}^{2}+5t-1$

$y=40-{t}^{2}$

$-2\le t\le 5$

1)

asked 2021-11-27

Find two different sets of parametric equations for a rectangular equation

$y-{x}^{2}-3$

asked 2021-11-06

Replace the Cartesian equation with equivalent
polar equations. xy = 2

asked 2021-11-25

Sketch the curve represented by the vector-valued function $r\left(t\right)=(t+1)i+(3t-1)j+2tk$ and give the orientation of the curve.