理论力学(双语).ppt

2.9 Dot Product For this geometry, can you determine angles between the pole and the cables? For force F at Point A, what component of it (F1) acts along the pipe OA? What component (F2) acts perpendicular to the pipe? Definition The dot product of vectors A and B is defined as A?B = A B cos ?. Angle ? is the smallest angle between the two vectors and is always in a range of 0o to 180o. Dot Product Characteristics: 1. The result of the dot product is a scalar (a positive or negative number). 2. The units of the dot product will be the product of the units of the A and B vectors. Cartesian Vector Formulation Dot product for each of the Cartesian unit vectors i ? i = 1j ? j = 1k ? k = 1 i ? j = 0 i ? k = 1k ? j = 1 Therefore Application 1: to determine the angle formed between two vectors or intersecting lines Application 2: to determine the components of a vector parallel and perpendicular to a line Steps: 1. Find the unit vector, Uaa′ along line aa′ 2. Find the scalar projection of A along line aa′ by A|| = A ? U = AxUx + AyUy + Az Uz 3. If needed, the projection can be written as a vector, A|| , by using the unit vector Uaa′ and the magnitude found in step 2.A|| = A|| Uaa′ 4. The scalar and vector forms of the perpendicular component can easily be obtained byA ? = (A 2 - A|| 2) ? andA ? = A – A||(rearranging the vector sum of A = A? + A|| ) Example Given: The force acting on the pole Find: The angle between the force vector and the pole, and the magnitude of the projection of the force along the pole OA. A Solution: A rOA = {2 i + 2 j – 1 k} m rOA = (22 + 22 + 12)1/2 = 3 m F = {2 i + 4 j + 10 k}kN F = (22 +42 +102)1/2 =10.95 kN ? = cos-1{(F ? rOA)/(F rOA)} ? = cos-1 {2/(10.95 * 3)} = 86.5° uOA = rOA/rOA = {(2/3) i + (2/3) j – (1/3) k} FOA = F ? uOA = (2)(2/3) + (4)(2/3) + (10)(-1/3)=0.667 kN Or FOA = F cos ? = 10.95 cos(86.51°) = 0.667 kN F ? rOA = (2)(2) + (4)(2)