Understanding the Dot Product: Determining Angles and Projections, Lecture notes of Geometry

Download Understanding the Dot Product: Determining Angles and Projections and more Lecture notes Geometry in PDF only on Docsity! DOT PRODUCT Objective: Students will be able to use the dot product to: a) determine an angle between two vectors, and, b) determine the projection of a vector along a specified line. APPLICATIONS 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? USING THE DOT PRODUCT TO DETERMINE THE ANGLE BETWEEN TWO VECTORS If we know two vectors in Cartesian form, finding  is easy since we have two methods of doing the dot product. ABcos = + cos = + cos = + More usually just written orcos = · cos = · 2D Example – Find  4 ̂ 7 ̂ 10 2 First ̂ ̂ and ̂ ̂ cos = · = + = So  = 48.95° Projection If we divide both sides in the of the definition above by the magnitude B, we can get the magnitude of the projection of in that direction. A dot product finds how much of is in the same direction as and then multiplies it by the magnitude of B Note! Vectors are not fixed to a point is space. Can do dot product on two vectors that are not touching and can find the angle between them and any projection of one vector along the other.  What if B is in the other direction? Vectors are not fixed to a point is space. Can do dot product on two vectors that are not touching and can find the angle between them and any projection of one vector along the other.  2D Example – Find B and  4 ̂ 7 ̂ 10 2 First ̂ ̂.  4 ̂ 7 ̂ · ̂ ̂ ̂ ̂ + ̂ ̂ 540 104 ̂ 108 104 ̂  4 ̂ 7 ̂ 540 104 ̂ 108 104 ̂ 124 104 ̂ 624 104 ̂ CONCEPT QUIZ 1. If a dot product of two non-zero vectors is 0, then the two vectors must be _____________ to each other. A) parallel (pointing in the same direction) B) parallel (pointing in the opposite direction) C) perpendicular D) cannot be determined. 2. If a dot product of two non-zero vectors equals -1, then the vectors must be ________ to each other. A) parallel (pointing in the same direction) B) parallel (pointing in the opposite direction) C) perpendicular D) cannot be determined. Example Given: The force acting on the pole. Find: The angle between the force vector and the pole, the magnitude of the projection of the force along the pole AO, as well as FAO (F||) and F.Plan: 1. Get rAO 2.  = cos-1{(F • rAO)/(F rAO)} 3. FAO = F • uAO or F cos  4. FAO = F|| = FOA uAO and F = F  F|| rAO = {-3 i + 2 j – 6 k} ft. rAO = (32 + 22 + 62)1/2 = 7 ft. F = {-20 i + 50 j – 10 k}lb F = (202 + 502 + 102)1/2 = 54.77 lb  = cos-1{(F • rAO)/(F rAO)}  = cos-1 {220/(54.77 × 7)} = 55.0° F • rAO = (-20)(-3) + (50)(2) + (-10)(-6) = 220 lb·ft uAO = rAO/rAO = {(-3/7) i + (2/7) j – (6/7) k} FAO = F • uAO = (-20)(-3/7) + (50)(2/7) + (-10)(-6/7) = 31.4 lb Or FAO = F cos  = 54.77 cos(55.0°) = 31.4 lb