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Notes

\(\text{Equation of a Plane Given 3 Points}\)

\( \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)

Practice Problems

Are the following points coplanar?

\(\textbf{1)}\) \((1,2,3), (4,2,6), (-2,8,9), \&\, (3,4,8)\)

Are the Points Coplanar?

\(\text{Yes, the 4 points are coplanar}\)

Show Work

\(\,\,\,\text{Find the equation of the plane through the first 3 points}\)

\(\,\,\,\,\,\,\,\,\,\,\,\,\left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,\left| {\begin{array}{ccc}x-1 & y-2 & z-3 \\4-1 & 2-2 & 6-3 \\-2-1 & 8-2 & 9-3 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,\left| {\begin{array}{ccc}x-1 & y-2 & z-3 \\3 & 0 & 3 \\-3 & 6 & 6 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,(x-1)(0 \cdot 6-3 \cdot 6) – (y-2)(3 \cdot 6-3 \cdot (-3)) + (z-3)(3 \cdot 6-0 \cdot (-3)) = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,(x-1)(-18) – (y-2)(27) + (z-3)(18) = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,-18x + 18 – 27y + 54 + 18z -54 = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,-18x – 27y + 18z = -18\)

\(\,\,\,\text{The equation of the plane is } 2x+3y-2z=2 \)

\(\,\,\,\text{Plug the 4th point in to the equation to see if it is on the plane}\)

\(\,\,\,\,\,\,\,\,\,\,\,\,2(3)+3(4)-2(8)=2 \)

\(\,\,\,\,\,\,\,\,\,\,\,\,6+12-16=2 \)

\(\,\,\,\,\,\,\,\,\,\,\,\,2=2 \)

\(\,\,\,\text{Therefore, the 4 points are coplanar}\)

\(\textbf{2)}\) \((5,3,2), (7,7,7), (-1,0,5), \&\, (1,8,9)\)

Are the Points Coplanar?

\(\text{No, the 4 points are not coplanar}\)

Show Work

\(\,\,\,\text{Find the equation of the plane through the first 3 points}\)

\(\,\,\,\,\,\,\,\,\,\,\,\, \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\, \left| {\begin{array}{ccc}x-5 & y-3 & z-2 \\7-5 & 7-3 & 7-2 \\-1-5 & 0-3 & 5-2 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\, \left| {\begin{array}{ccc}x-5 & y-3 & z-2 \\2 & 4 & 5 \\-6 & -3 & 3 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,(x-5)(4 \cdot 3-5 \cdot (-3)) – (y-3)(2 \cdot 3-5 \cdot (-6)) + (z-2)( 2 \cdot (-3)-4 \cdot (-6)) = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,(x-5)(27) + (y-3)(-36) + (z-2)(18) = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,27x – 135 – 36y + 108 + 18z -36 = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,27x-36y+18z-63 = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,3x-4y+2z=7 \)
\(\,\,\,\text{The equation of the plane is } 3x-4y+2z=7 \)

\(\,\,\,\text{Plug the 4th point in to the equation to see if it is on the plane}\)

\(\,\,\,\,\,\,\,\,\,\,\,\,3(1)-4(8)+2(9)\ne7\)

\(\,\,\,\,\,\,\,\,\,\,\,\,3-32+18\ne7\)

\(\,\,\,\,\,\,\,\,\,\,\,\,-11\ne7\)

\(\,\,\,\text{Therefore, the 4 points are not coplanar}\)

\(\textbf{3)}\) \((2,2,2), (4,-1,-1), (0,0,1), \&\, (-4,1,3)\)

Are the Points Coplanar?

\(\text{Yes, the 4 points are coplanar}\)

Show Work

\(\,\,\,\text{Find the equation of the plane through the first 3 points}\)

\(\,\,\,\\,\,\,\,\,\,,\,\, \left| {\begin{array}{ccc}x-x_1 & y-y_1 & z-z_1 \\x_2-x_1 & y_2-y_1 & z_2-z_1 \\x_3-x_1 & y_3-y_1 & z_3-z_1 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\, \left| {\begin{array}{ccc}x-2 & y-2 & z-2 \\4-2 & -1-2 & -1-2 \\0-2 & 0-2 & 1-2 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\, \left| {\begin{array}{ccc}x-2 & y-2 & z-2 \\2 & -3 & -3 \\-2 & -2 & -1 \\\end{array} } \right|=0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,(x-2)(-3 \cdot (-1)-(-3) \cdot (-2)) – (y-2)(2 \cdot (-1)-(-3) \cdot (-2)) + (z-2)(2 \cdot (-2)-(-3) \cdot (-2)) = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\,(x-2)(-3) – (y-2)(8) + (z-2)(-10) = 0\)

\(\,\,\,\,\,\,\,\,\,\,\,\, -3x+6 -8y +16 -10z+20 = 0\)

\(\,\,\,\text{The equation of the plane is } 3x-8y+10z=10 \)

\(\,\,\,\text{Plug the 4th point in to the equation to see if it is on the plane}\)

\(\,\,\,\,\,\,\,\,\,\,\,\,3(-4)-8(1)+10(3)=10\)

\(\,\,\,\,\,\,\,\,\,\,\,\,-12-8+30=10\)

\(\,\,\,\,\,\,\,\,\,\,\,\,10=10\)

\(\,\,\,\text{Therefore, the 4 points are coplanar}\)

\(\textbf{4)}\) \((0,0,0), (1,1,1), (7,8,9), \&\, (5,8,3)\)

Are the Points Coplanar?

\( \text{No, the 4 points are not coplanar.}\)

\(\textbf{5)}\) \((1,2,3), (1,4,6), (1,8,-3), \&\, (0,14,-9)\)

Are the Points Coplanar?

\( \text{Yes, the 4 points are coplanar} \)

See Related Pages\(\)

\(\bullet\text{ Displacement Vectors}\)
\(\,\,\,\,\,\,\,\,(x_2-x_1)\vec{i}+(y_2-y_1)\vec{j}…\)

\(\bullet\text{ Magnitude, Direction, and Unit Vectors}\)
\(\,\,\,\,\,\,\,\,|\vec{u}|=\sqrt{a^2+b^2}…\)

\(\bullet\text{ Dot Product}\)
\(\,\,\,\,\,\,\,\,a \cdot b=x_1 x_2+ y_1 y_2…\)

\(\bullet\text{ Parallel and Perpendicular Vectors}\)
\(\,\,\,\,\,\,\,\,⟨8,2⟩ \text{ and } ⟨−4,−1⟩…\)

\(\bullet\text{ Scalar and Vector Projections}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{a \cdot b}{|b|^2} \, \vec{b}…\)

\(\bullet\text{ Cross Product}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Equation of a Plane}\)
\(\,\,\,\,\,\,\,\,Ax+By+Cz=D…\)

In Summary

Coplanar points are a group of points that lie on the same plane in space. These points can be represented by coordinates or equations. In order to determine if a group of points are coplanar, we make sure the equation of the plane is true when the coordinates are plugged in.

The concept of coplanar points is introduced in high school geometry classes, where verification is usually done by looking at an image of the points. In more advanced linear algebra courses students are able to show the points are coplanar by working with matrices.

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