Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 45Moreover, if we assume that the object moves along a line from point P to point Q, thend = ‖ −→ PQ‖, so that the work can be expressed entirely in vector form asW = ‖F‖‖ −→ PQ‖The vector −→ PQ is called the displacement vector for the object.In the case where a constant force F is not in the direction of motion, but rather makesan angle θ with the displacement vector, then we define the work W done by F to beW = (‖F‖cosθ)‖ −→ PQ‖ = F·−→ PQ (3.13)•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.