Three Dimensional Geometry Lecture 8

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Lecture - 8 Chapter 11 Three Dimensional Geometry

“Whenever you find yourself on the side of the majority, it is time to pause and reflect.” –Mark Twain

In this video, I am discussing equation of plane in three dimensional geometry and its derivation.

If One passing point and a normal vector is given then equation of plane in both the vector form and the Cartesian form:
\((\vec{r}-\vec{a})\cdot\vec{n}=0 \)
\((x-x_1)a+(y-y_1)b+(z-z_1)c=0\)

If three non-collinear point are given then equation of plane in both the vector form and the Cartesian form:

\((\vec{r}-\vec{a})\cdot[(\vec{b}-\vec{a})\times(\vec{c}-\vec{a})]=0\)
\(\left|\begin{matrix}x-x_1&y-y_1&z-z_1\\x_2-x_1&y_2-y_1&z_2-z_1\\x_3-x_1&y_3-y_1&z_3-z_1\\\end{matrix}\right|=0\)

If intercepts on the axes are given then equation of plane in the Cartesian form:
Intercept on x-axis: A
Intercept on y-axis: B
Intercept on z-axis: C
We can use identity for three passsing points to find equation of plane.
Equation of plane: \(\frac{x}{A}+\frac{y}{B}+\frac{z}{C}=1\)

6. The foot of perpendicular drawn from origin to the plane is (4, −2, −5). Find the equation of the plane.

17. Write the equation of the plane whose intercepts on the coordinate axes are 2, -3 and 4.

18. Reduce the equations of the following planes in intercept form and find its intercepts on the coordinate axes:
a. 4x+3y-6z-12=0
b. 2x+3yz=6
c. 2xy+z=5

19. Find the equation of a plane which meets the axes in A, B and C, given that the centroid of the triangle ABC is the point \((\alpha,\beta,\gamma)\) .

20. Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.

21. A plane meets the coordinate axes at A, B and C respectively such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.

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