CHAPTER 2 THE DOT CROSS PRODUCTS AND sin  Thus area  A B  A B We can also deduce from this that the area of a triangle is half the area of a parallelogram. Hence area of a triangle is . A 1 B 2 Example 2.14 Find the area of the triangle having vertices at P(1,3,2), Q(2,-1,1), R(-1,2,3). We have 4 ; , and    PQ iˆ kˆ PR kˆ ˆj ˆj area of triangle 1   PQ PR 2 2  1 107 2 TRIPLE PRODUCTS Scalar Triple Product: If are three vectors, the scalar formed by the product A, B, C   is called the scalar triple product. A If , . A  A1iˆ  A2  A3kˆ B  B1iˆ  B2  B3kˆ and C  C1iˆ  C2  C3kˆ ˆj ˆj ˆj iˆ kˆ ˆj Then A B  C  B1 C3 A2 C1 A3 B2 B  C  B1 C2 B3 B2 C1 C3 C2 A1 B3
CHAPTER 2 THE DOT AND CROSS PRODUCTS Thus area  A sin  B  A B We can also deduce from this that the area of a triangle is half the area of a parallelogram. Hence area of a triangle is 1 2 A B . Example 2.14 Find the area of the triangle having vertices at P(1,3,2), Q(2,-1,1), R(-1,2,3). We have PQ  iˆ 4 ˆj kˆ ; PR  ˆj  kˆ , and area of triangle  1 PQ  PR 2 2  1 107 2 TRIPLE PRODUCTS Scalar Triple Product: If A, B, C are three vectors, the scalar formed by the product A   is called the scalar triple product. If A  A1iˆ  A2 ˆj  A3kˆ , B  B1iˆ  B2 ˆj  B3kˆ and C  C1iˆ  C2 ˆj  C3kˆ . iˆ ˆj kˆ Then B  C  B1 C1 A1 A B  C  B1 C1 B2 C2 A2 B2 C2 B3 C3 A3 B3 C3