Lecture 13 Three Dimensional Geometry

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Lecture - 13 Chapter 11 Three Dimensional Geometry

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

11. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.

12. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX-plane.

13. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the XY-plane.

14. Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x+y+z=7 .

17. Find the distance of the point (-1,-5,-10) from the point of intersection of the line \(\vec{r}=2\hat{i}-\hat{j}+2\hat{k}+\lambda(3\hat{i}+4\hat{j}+2\hat{k})\) and the plane \(\vec{r}.(\hat{i}-\hat{j}+\hat{k})=5\) .

18. Find the coordinates of the point where the line through (3,-4,-5) and(2,-3,1) crosses the plane passing through three points (2,2,1) , (3,0,1) and (4,-1,0).

20. Prove that the line through A(0, −1, −1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(−4,4, 4).

21. Find the equation of plane which perpendicularly bisect the line joining the points A(2, 3, 4) and B(4, 5, 8).

22. If a line drawn from the point (-2,-1,-3) meets a plane at right angle at the point (1,-3,3) , find the equation of the plane.

23. Find the equations of the line passing through the point (3, 0, 1) and parallel to the planes x+2y=0 and 3y-z=0 .

24. Show that the lines \(\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}\) and \(\frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}\) intersect each other. Find the point of intersection.

25. A vector of magnitude 8 units is inclined to x-axis at 45° , y-axis at 60° and an acute angle with the z-axis. If a plane passes through a point \((\sqrt2,-1,1)\) and is normal to \( \vec{n}\) , find its equation in vector form.

26. Show that the line whose vector equation is \(\vec{r}=(2\hat{i}-2\hat{j}+3\hat{k})+\lambda(\hat{i}-\hat{j}+4\hat{k})\) parallel to the plane whose vector equation is \(\vec{r}.(\hat{i}+5\hat{j}+\hat{k})=5\) . Also, find the distance between them.

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