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Paper 1 · November 2010 · Vectors

For three non-zero vectors aa, bb and cc, the scalar triple product a⋅(b×c)a \cdot (b \times c) measures which quantity?

Athe area of the parallelogram on b and c
Bthe volume of the parallelepiped on a, b, c
Cthe volume of the tetrahedron on a, b, c
Dthe length of the projection of a onto b x c

Explanation

b×cb\times c is normal to the plane of bb and cc and has magnitude equal to the area of the parallelogram they span. Taking the scalar product with aa multiplies that area by the height of aa above the plane, which is the volume of the parallelepiped with aa, bb and cc as its edges. Here the volume is ∣36∣=36|36| = 36 cubic units, and because the value is not zero the three vectors are not coplanar.

Derived from ZIMSEC Further Mathematics 9187/1, November 2010, Q1

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