Three - Dimensional Geometry Examples and Solutions

Find the distance of the point (2, 12, 5) from the point of intersection of the line 

${\displaystyle \overrightarrow{r}=2\hat{i}-4\widehat{+}2\hat{k}+\lambda (3\hat{i}+4\hat{j}+2\hat{k})}$

Find the coordinates of the foot of perpendicular drawn from the point A (-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

If lines ${\displaystyle \frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} \text{and} \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}}$ intersect, then find the value of k and hence find the equation of the plane containing these lines.

Find the coordinates of the foot of perpendicular and perpendicular distance from the point P(4,3,2) to the plane x + 2y + 3z = 2. Also find the image of P in the plane.

Find the distance between the point (−1, −5, −10) and the point of intersection of the line ${\displaystyle \frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}}$ and the plane x-y+z=5

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

Show that the lines ${\displaystyle \frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7} \text{and} \frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}}$ intersect. Also find their point of intersection

Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle π/4 with the plane x + y = 3. Also find the equation of the plane