Write the Cartesian equation of the following line given in vector form: \(\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )\)

What are the direction cosines of a line, which makes equal angles with the coordinate axes?

Find the distance of the plane 3x - 4y + 12z = 3 from the origin.

Write the direction cosines of a line parallel to the z-axis.

If a lines makes angle 60° and 45° with the positive directions of x-axis and z-axis respectively, then find the angle that it makes with the y-axis.

Show that the line through the points (0, 3, 2), (3, 5, 6) is perpendicular of the line through the points (1, - 1, 2) and (3,4, - 2).

Find the vector equation of the plane which is at a distance of 5 units from the origin and normal to the plane is \(3\check { i } -2\check { j } +6\check { k } \)

Find the angle between line \(\frac { x-1 }{ 6 } =\frac { y+3 }{ 2 } =\frac { z-2 }{ 3 } \) and the plane 2x - Y + 2z - 13 = 0.

A plane meets the co-ordinate axes in A, B, and C such that the centroid of \(\triangle\)ABC is the point \((\alpha,\beta,\gamma).\) Show that the equation of the plane is \({{x}\over{\alpha}}+{{y}\over{\beta}}+{{z}\over{\gamma}}=3.\)

Find the direction cosines of the line passing through the two points (-2, 4, -5) and (1, 2, 3)

Find the direction-cosines of x, y and z-axis.

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.

Find the co-ordinates of the foot of the perpendicular and the perpendicular distance of the point (1, 3, 4) from the plane 2x - y + z + 3 = 0. Find also, the image of the point in the plane.

Find the equation of the plane which is perpendicular to the plane 5x+3y+6z+8=0 and which contains the line of intersection of the planes x + 2y + 3z - 4 = 0 and 2x + y - z + 5 = 0.

Find the point on the line \(\frac { x+2 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-3 }{ 2 } \) at a distance \(3\sqrt { 2 } \) from the point (1, 2, 3)

If lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 } \) and \(\frac { x-3 }{ 1 } =\frac { y-k }{ 32 } =\frac { z }{ 1 } \)intersect, then find the value of k and hence find the equation of plane containing these lines.

Find the equation of the plane that contains the point (1,-1,2) and is perpendicular to both the planes 2x + 3y - 2z = 5 and x + 2y - 3z = 8. Hence find the distance of point P(-2, 5, 5) from the plane obtained above.

Find the equation of the plane passing through the line of intersection of the plane \(\overrightarrow{r}.(\hat{i}+3\hat{j})-6=0\) and \(\overrightarrow{r}.(3\hat{i}-\hat{j}-4\hat{k})=0,\) which is at a unit distance from the origin.

Find the vector equation of the plane that contains the lines \(\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+2\hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-2\hat{k}).\)
Also, find the length of perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.