1. Find the volume of parallelepiped whose edges are i + j + k , i - j + k and i + j - k
Sol: Given edges i + j + k
i - j + k
i + j - k
1(1 - 1) - 1(-1 - 1) + 1 (1 + 1)
= 4 Cubic unit
2. Find the volume of the tetrahedron with edges i + j + k, i - j + k and i + 2j - k
Sol: Given edges i + j + k
i - j + k
i + 2j - k
3. Find the volume of tetrahedron with vertices (1, 1, 3) (4, 3, 2), (5, 2, 7) and (6, 4, 8).
Sol: Given vertices
4. Find the perpendicular distance from the origin to the plane passing through the points
(1, -2, 5), (0, -5, -1) and (-3, 5, 0).
Sol: Given 
= (1, -2, 5) = i - 2j + 5k
= (0, -5, -1) = - 5j - k
= (-3, 5, 0) = - 3i + 5j
5. Find the cartesian equation of the plane passing through the point 
= (0, 2, 1) and parallel to the vectors 
= (1, 2, 3) and 
= (1, 3, 2).
Sol: Given 
= (0, 2, 1) = 2j + k
= (1, 2, 3) = i + 2j + 3k
= (1, 3, 2) = i + 3j + 2k
Let 
= (x, y, z) = xi + yj + zk be any point on π
Now, 
- 
= xi + (y - 2) j + (z - 1) k
6. Find the vector equation of the plane passing through three non-collinear points -2i + 6j - 6k, -3i + 10j - 9k, - 5i - 6k. Also find its cartesian equation
7. Find the equation of the plane passing through the points 3i - 5j - k, -i + 5j + 7k and parallel to 3i - j + 7k
Sol: Given 
= 3i - 5j - k
= -i + 5j + 7k
= 3i - j + 7k
Let 
= xi + yj + zk be any point on π
- 
= (x - 3) i + (y + 5) j + (z + 1) k
- 
= -2i + 5j + 4k
Writer: Sayyad Anwar
