# 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.