Question

Find an equation of the plane. The plane through the points (2, 1, 2), (3, −8, 6), and (−2, −3, 1)

Right triangles and trigonometry
ANSWERED
asked 2021-05-31
Find an equation of the plane. The plane through the points (2, 1, 2), (3, −8, 6), and (−2, −3, 1)

Answers (1)

2021-06-01

Let the given points be P(2,1,2),Q(3,-8,6), and R(-2,-3,1)
\(\vec{PQ}=\langle3-2,-8-1,6-2\rangle=\langle1,-9,4\rangle\)
\(\vec{PR}=\langle-2-2,-3-1,1-2\rangle=\langle-4,-4,-1\rangle\)
\(\vec{PQ}\times\vec{PR}=\begin{vmatrix}i&j&k\\1&-9&4\\-4&-4&-1\end{vmatrix}=(9+16)i+(-16+1)j+(-4-36)k\)
\(=25i-15j-40k\)
Therefore, the normal vector to the plane is
\(\vec{n}=\langle25,-15,-40\rangle\)
Since, the plane passes through all the three points we can choose any point to find its equation. So, the equation of the plane through the point P(2,1,2) with normal vector \(\vec{n}=\langle25,-15,-40\rangle\) is
\(25(x-2)-15(y-1)-40(z-2)=0\)
\(\Rightarrow 25x-50-15y+15-40z+80=0\)
\(\Rightarrow 25x-15y-40z+45=0\)

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-06-03
Find an equation of the plane.
The plane through the points (2,1,2), (3,-8,6), and (-2,-3,1)
asked 2021-06-04
Find an equation of the plane.
The plane through the points (4, 1, 4), (5, -8, 6), and (-4, -5, 1)
asked 2021-03-06

Find an equation of the plane passing through the three points given. \(P = (2, 0, 0),\)

\(Q = (0, 4, 0),\)

\(R = (0, 0, 2)\)

asked 2021-05-16
Find an equation for the plane that
(a) is perpendicular to \(v=(1,1,1)\) and passes through (1,0,0).
(b) is perpendicular to \(v=(1,2,3)\) and passes through (1,1,1)
(c) is perpendicular to the line \(l(t)=(5,0,2)t+(3,-1,1)\) and passes through (5,-1,0)
(d) is perpendicular to the line \(l(t)=(-1,-2,3)t+(0,7,1)\) and passes through (2,4,-1).
asked 2021-06-18

find an equation of the form \(y=ax^2 + bx + c\), whose graph passess through the points (2, 3), (-2, 7) and (1, -2). You must use a system of linear equations in three variables for this problem.

asked 2021-05-03
Find a nonzero vector orthogonal to the plane through the points P, Q, and R, P(0,-2,0), Q(4,1,-2), R(5,3,1)
asked 2021-06-12

For the following exercises, use the given information about the polynomial graph to write the equation. Degree 3. Zeros at \(x = -3, x = -2\ and\ x = 1\). y-intercept at (0, 12).

asked 2021-05-14
Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR
Consider the points below
P(1,0,1) , Q(-2,1,4) , R(7,2,7)
a) Find a nonzero vector orthogonal to the plane through the points P,Q and R
b) Find the area of the triangle PQR
asked 2021-05-18
Find an equation of the tangent plane to the given surface at the specified point.
\(z=2(x-1)^2+6(y+3)^2+4,\ (3,-2,18)\)
asked 2021-06-07
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
A(1,0,-1), B(5,-3,0), C(1,5,2)
Find \(\angle CAB,\angle ABC, \angle BCA\)
...