Three Dimensional Geometry Lecture 9

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Lecture - 9 Chapter 11 Three Dimensional Geometry

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

7. Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes 2x+3y-2z=5and x+2y-3z=8 .

8. Find the equation of the plane passing through (a,b,c) and parallel to the plane \(\vec{r}.(\hat{i}+\hat{j}+\hat{k})=2\).

9. Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z=0 .

10. Find the equation of the plane passing through the points (1, −1, 2) and (2, −2, 2) and which is perpendicular to the plane 6x-2y+2z=9.

11. Find the equation of the plane through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x+2y+2z=5.

15. In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them:
a. 7x+5y+6z+30=0,3x-y-10z+4=0
b. 2x+y+3z-2=0,x-2y+5=0
c. 2x-2y+4z+5=0,3x-3y+6z-1=0
d. \(\vec{r}.(2\hat{i}+2\hat{j}-3\hat{k})=5,\vec{r}.(3\hat{i}-3\hat{j}+5\hat{k})=3\)
e. \(\vec{r}.(2\hat{i}-3\hat{j}+4\hat{k})=1,\vec{r}.(-\hat{i}+\hat{j})=4\)
f. \(\vec{r}.(2\hat{i}-\hat{j}+2\hat{k})=6,\vec{r}.(3\hat{i}+6\hat{j}-2\hat{k})=9\)

16. Determine the value of \lambda for which the following planes are perpendicular to each other
a. \(\vec{r}.(\hat{i}+2\hat{j}+3\hat{k})=7\),\(\vec{r}.(\lambda\hat{i}+2\hat{j}-7\hat{k})=26\)
b. 2x-4y+3z=5,\(x+2y+\lambda\ z=5\)
c. 3x-6y-2z=7,\(2x+y-\lambda\ z=5\)

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