Two motorcycles A and B are running at the speed more than allowed speed on the road along the lines \(\vec{r}=\lambda(\hat{i}+2 \hat{j}-\hat{k}) \text { and } \vec{r}=3 \hat{i}+3 \hat{j}+\mu(2 \hat{i}+\hat{j}+\hat{k})\), respectively.

Based on the above information, answer the following questions.
(i) The cartesian equation of the line along which motorcycle A is running, is

(ii) The direction cosines of line along which motorcycle A is running, are

(iii) The direction ratios of line along which motorcycle B is running, are

(iv) The shortest distance between the gives lines is

(v) The motorcycles will meet with an accident at the point

Consider the following diagram, where the forces in the cable are given.

Based on the above information, answer the following questions.
(i) The cartesian equation of line along EA is

(ii) The vector \(\overline{E D}\) is

(iii) The length of the cable EB is

(iv) The length of cable EC is equal to the length of

(v) The sum of all vectors along the cables is

Suppose the floor of a hotel is made up of mirror polished Kota stone. Also, there is a large crystal chandelier attached at the ceiling of the hotel. Consider the floor of the hotel as a plane having equation x - 2y + 2z = 3 and crystal chandelier at the point (3, -2, 1).

Based on the above information, answer the following questions.
(i) The d.r's of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

(ii) The length of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

(iii) The equation of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

(iv) The equation of plane parallel to the plane x - 2y + 2z = 3, which is at a unit distance from the point (3, -2, 1) is

(v) The image of the point (3, -2, 1) in the given plane is

In a diamond exhibition, a diamond is covered in cubical glass box having coordinates 0(0, 0, 0), A(1, 0, 0), B(1, 2, 0), C(0, 2, 0), O'(0,0,3), A'(1, 0, 3), B'(1, 2, 3) and C(0, 2, 3).

Based on the above information, answer the following questions.
(i) Direction ratios of OA are

(ii) Equation of diagonal OB' is

(iii) Equation of plane OABC is

(iv) Equation of plane O' A' B' C is

(v) Equation of plane ABB' A' is

The equation of motion of a rocket are: x = 2t, y = -4t, z = 4t, where the time 't' is given in seconds, and the distance measured is in kilometres.

Based on the above information, answer the following questions.
(i) What is the path of the rocket?

(ii) Which of the following points lie on the path of the rocket?

(iii) At what distance will the rocket be from the starting point (0, 0, 0) in 10 seconds?

(iv) If the position of rocket at certain instant of time is (3, -6, 6), then what will be the height of the rocket from the ground, which is along the xy-plane?

(v) At certain instant of time, if the rocket is above sea level, where equation of surface of sea is given by 3x - y + 4z = 2 and position of rocket at that instant of time is (1, -2,2), then the image of position of rocket in the sea is