CHAPTER 2 THE DOT CROSS PRODUCTS AND B 9 i 7kˆ Thus 2 4  k  j ˆj  bˆ   4iˆ B  and j 19 9 4 k  9 9  Example 2.8 2 cos Prove that C  A2  B AB where , , are the three sides of a triangle A B C 2 2 with  the angle between the vectors . A and B C B  A From the diagram we have or    A B C C A B     Thus C C 2   A A A B B B   2 cos C A2 B AB Hence 2 2 2 cos We call C  A2  B AB the Cosine rule for plane triangles. 2 2 Example 2.9 Find an equation for the plane perpendicular to the vector  2iˆ  3  6kˆ and A ˆj passing through the terminal point of the vector 5 3kˆ . Also find the B  iˆ   ˆj distance from the origin to the plane. Z P(x,y,z) Q A r
CHAPTER 2 THE DOT AND CROSS PRODUCTS Thus bˆ  B B  4iˆ 4 ˆj  7kˆ 2  i j  9 k and 4 9 j  k  9 19 9 Example 2.8 Prove that C 2  A2  B 2 2 AB cos where A , B , C are the three sides of a triangle with  the angle between the vectors A and B . C B  A From the diagram we have A  B  C or C  A B Thus Hence C C      A A 2 A B  B B C 2  A2  B 2 2 AB cos We call C 2  A2  B 2 2 AB cos the Cosine rule for plane triangles. Example 2.9 Find an equation for the plane perpendicular to the vector A  2iˆ  3 ˆj  6kˆ and passing through the terminal point of the vector B  iˆ  5 ˆj  3kˆ . Also find the distance from the origin to the plane. Z P(x,y,z) Q A r 