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Consider the following two statements:
Statement 1: If A, B and C are points with position vectors a = 2î + ĵ + k̂, b = 3î - ĵ + 3k̂ and c = î + 7ĵ - 5k̂, then the figure OABC forms a tetrahedron.
Statement 2: If the position vectors a, b and c of points A, B and C are non-coplanar, then OABC is a tetrahedron, where O denotes the origin.
Choose the correct option.
- Statement 1 is true, Statement 2 is true, and Statement 2 correctly explains Statement 1
- Statement 1 is true, Statement 2 is true, but Statement 2 does not correctly explain Statement 1
- Statement 1 is false, Statement 2 is true
- Statement 1 is true, Statement 2 is false
Correct answer: Statement 1 is true, Statement 2 is true, and Statement 2 correctly explains Statement 1
Solution
The scalar triple product [a b c] equals the determinant of rows (2,1,1),(3,-1,3),(1,7,-5) = 2(5-21)-1(-15-3)+1(21+1) = 8, which is nonzero. So a,b,c are non-coplanar, OABC is a tetrahedron (Statement 1 true), and Statement 2 is the correct general reason.
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