Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 46•P‖F‖FWork = ‖F‖‖ −→ PQ‖•Q•P‖F‖Fθ‖F‖cosθWork = (‖F‖cosθ)‖ −→ PQ‖•QExample 3.19 A force F = 8i + 5j in pound moves an object from P(1,0) to Q(7,1),distance measured in feet. How much work is done?Solution .........Example 3.20 A wagon is pulled horizontally by exerting a constant force of 10lb on thehandle at an angle of 60 ◦ with the horizontal. How much work is done in moving the wagon50 ft?Solution .........3.4 Cross ProductDeterminantsBefore we define the cross product, we need to define the notion of determinant.Definition 3.4 The determinant of a 2×2 matrix of real number is defined by∣ a ∣1 a 2∣∣∣= ab 1 b 1 b 2 −a 2 b 1 .2Definition 3.5 The determinant of a 3×3 matrix of real number is defined as a combinationof three 2×2 determinants, as follows:∣ a 1 a 2 a 3∣∣∣∣∣∣ ∣ ∣ ∣ ∣ ∣ ∣∣∣ b b 1 b 2 b 3 = a 2 b 3∣∣∣∣∣∣ b1 −a 1 b 3∣∣∣∣∣∣ b∣ cc 1 c 2 c 2 c 2 +a 1 b 2∣∣∣3 c 1 c 3 (3.14)3 c 1 c 23Note that Equation (3.14) is referred to as an expansion of the determinant alongthe first row.Cross ProductWe now turn to the main concept in this section.Definition 3.6 If u = 〈u 1 ,u 2 ,u 3 〉 and v = 〈v 1 ,v 2 ,v 3 〉 are vectors in 3-space, then thecross product u×v is the vector defined byu×v =∣ u ∣ 2 u 3∣∣∣i−v 2 v 3∣ u ∣ 1 u 3∣∣∣j+v 1 v 3∣ u ∣1 u 2∣∣∣k (3.15)v 1 v 2