Question
Let vectors a, b, and c be such thata · b = 0, b · c
= 0, and c · a = 0. If |a| = 2, |b| = 3, |c| = 6, then the volume of the parallelepiped formed is:Solution
We are given that:
- a · b = 0 → vectors a and b are perpendicular
- b · c = 0 → vectors b and c are perpendicular
- c · a = 0 → vectors c and a are perpendicular
Hence, they form a rectangular parallelepiped (i.e., a cuboid), and the volume is given by: Volume = |a · (b × c)| But when vectors are mutually perpendicular, the volume simplifies to: Volume = |a| × |b| × |c| Now substitute: |a| = 2
|b| = 3
|c| = 6 So, Volume = 2 × 3 × 6 = 36
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