ISC Class 12th Mathematics Chapter 11 - Three - dimensional Geometry Important Questions with Answers

A mobile tower is situated at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with three cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at point P(2, 3, 1) as shown in the figure below. The foot of the perpendicular from the point P on the plane is at the point `Q(43/29, 77/29, 9/29)`.

Answer the following questions.

1. Find the equation of the plane containing the points A, B and C.
2. Find the equation of the line PQ.
3. Calculate the height of the tower.

Show that the line whose vector equation is `vecr = (2hati - 2hatj + 3hatk) + ?(hati - hatj + 4hatk)` is parallel to the plane whose vector equation is `vecr.(hati + 5hatj + hatk) = 5`. Also find the distance between them.

Find the equation of the plane containing the line `x/(-2) = (y - 1)/3 = (1 - z)/1` and the point (–1, 0, 2).

 Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Write the sum of intercepts cut off by the plane `vecr.(2hati+hatj-k)-5=0` on the three axes

Find the value of p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l₁.

The Cartesian equation of the line is 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, –5, 0) and parallel to the given line.

Find the equation of the plane through the intersection of the planes `vecr.(hati + 3hatj - hatk) = 9` and `vecr.(2hati - hatj + hatj) = 3` and passing through the origin.

Find the image of a point having the position vector: `3hati - 2hatj + hat k` in the plane `vec r.(3hati - hat j + 4hatk) = 2`