Vardaan Learning Institute

Find the direction cosines of the line which is perpendicular to the lines with direction ratios $1, -2, -2$ and $0, 2, 1$.

Find the vector and Cartesian equations of the line passing through the point $(1, 2, -4)$ and perpendicular to the two lines: $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}$.

Find the angle between the lines whose direction cosines are given by the equations $l+m+n=0$ and $l^2+m^2-n^2=0$.

Find the angle between the lines represented by $2x = 3y = -z$ and $6x = -y = -4z$.

Find the vector equation of a line passing through $(2, -1, 1)$ and parallel to the line joining the points $(1, 2, 3)$ and $(-1, 4, 1)$.

Find the Cartesian equation of a line passing through $(2, -1, 3)$ and equally inclined to the positive coordinate axes.

Show that the line joining the origin to $(2, 1, 1)$ is perpendicular to the line determined by the points $(3, 5, -1)$ and $(4, 3, -1)$.

If two lines have direction ratios proportional to $a, b, c$ and $b-c, c-a, a-b$ respectively, find the angle between them.

Find the Cartesian equation of the line passing through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.

Find the angle between the pair of lines whose direction ratios are $(1, 1, 2)$ and $(\sqrt{3}-1, -\sqrt{3}-1, 4)$.

Calculate the shortest distance between the skew lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$.

Find the shortest distance between the parallel lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(2\hat{i} + 3\hat{j} + 4\hat{k})$.

Find the coordinates of the foot of the perpendicular and the image of the point $(1, 6, 3)$ with respect to the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$.

Find the shortest distance between the lines given by vector equations $\vec{r} = (\hat{i} - 2\hat{j} + 3\hat{k}) + t(-\hat{i} + \hat{j} - 2\hat{k})$ and $\vec{r} = (\hat{i} - \hat{j} + \hat{k}) + s(\hat{i} + 2\hat{j} + 2\hat{k})$.

Find the length and the coordinates of the foot of the perpendicular drawn from the point $(2, -1, 5)$ to the line $\frac{x-11}{10} = \frac{y+2}{-4} = \frac{z+8}{-11}$.

Find the image of the point $(5, 8, 15)$ in the line given by $\vec{r} = (6\hat{i} + 7\hat{j} + 7\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k})$.

Prove that the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-4}{5} = \frac{y-1}{2} = z$ intersect. Find their point of intersection.

Find the shortest distance between the lines passing through $(1,2,3)$ and parallel to $2\hat{i}+3\hat{j}+4\hat{k}$, and the line passing through $(2,4,5)$ parallel to $3\hat{i}+4\hat{j}+5\hat{k}$.

Find the shortest distance between the lines given by $\frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}$ and $\frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-7}{1}$.

A line passes through $(2, -1, 3)$ and is perpendicular to the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$. Obtain its vector equation.

Find the equation of the plane passing through the points $(2, 1, -1)$ and $(-1, 3, 4)$ and perpendicular to the plane $x - 2y + 4z = 10$.

Find the equation of the plane passing through the line of intersection of the planes $x + y + z = 1$ and $2x + 3y + 4z = 5$ which is perpendicular to the plane $x - y + z = 0$.

Find the equation of the plane passing through the points $(1, 1, -1), (6, 4, -5)$ and $(-4, -2, 1)$.

Find the equation of the plane passing through the line of intersection of the planes $2x + y - z = 3$ and $5x - 3y + 4z + 9 = 0$, and parallel to the line $\frac{x-1}{2} = \frac{y-3}{4} = \frac{z-5}{5}$.

Find the Cartesian equation of the plane passing through $(1, 2, 3)$ and perpendicular to the planes $x - 2y + z = 1$ and $2x + y + 3z = 2$.

A plane meets the coordinate axes at $A, B, C$ such that the centroid of $\Delta ABC$ is the point $(a, b, c)$. Show that the equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 3$.

Find the $x$, $y$, and $z$ intercepts of the plane passing through $(2, 3, 4)$ and parallel to the plane $3x - 4y + 5z = 6$.

Find the equation of the plane which contains the line of intersection of the planes $\vec{r} \cdot (\hat{i} + 2\hat{j} + 3\hat{k}) - 4 = 0$ and $\vec{r} \cdot (2\hat{i} + \hat{j} - \hat{k}) + 5 = 0$, and which is perpendicular to the plane $\vec{r} \cdot (5\hat{i} + 3\hat{j} - 6\hat{k}) + 8 = 0$.

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

Find the equation of the plane containing the line $\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z+2}{1}$ and the point $(0, 7, -7)$.

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 $\vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 5$.

Show that the lines $\frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}$ and $\frac{x+1}{-1} = \frac{y-2}{2} = \frac{z-5}{5}$ are coplanar. Also find the equation of the plane containing them.

Find the distance of the point $(1, -2, 3)$ from the plane $x - y + z = 5$ measured parallel to the line $\frac{x}{2} = \frac{y}{3} = \frac{z}{-6}$.

Find the coordinates of the image of the point $(1, 3, 4)$ in the plane $2x + y + z = 18$.

Find the distance of the point $(2, 3, 4)$ from the plane $3x + 2y + 2z + 5 = 0$ measured parallel to the line $\frac{x+3}{3} = \frac{y-2}{6} = \frac{z}{2}$.

Show that the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and the plane $x - 2y + z = 10$ are parallel. Find the perpendicular distance between them.

Find the equation of the plane containing the two parallel lines $\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{3}$ and $\frac{x}{2} = \frac{y-2}{-1} = \frac{z+1}{3}$.

Find the angle between the line $\frac{x-2}{3} = \frac{y+1}{-1} = \frac{z-3}{2}$ and the plane $3x + 4y + z + 5 = 0$.

Find the length and the foot of the perpendicular drawn from the point $(7, 14, 5)$ to the plane $2x + 4y - z = 2$.

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 $3x + 3y + z = 0$.