Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 Dot Product (Geometric Definition) Def.: The dot product of to vectors v and n in is given by here θ, satisfying 0, is the angle beteen v and. v Exercise a: Determine the dot product of each of the folloing pairs of vectors. (i) (ii) (iii) v v v v v v v v v (iv) (v) Observation : If then 0 v, Hint: θ 49.3987 v v v 5 7 cos(49.3987 ) Hint: θ 5.83765 v v v Observation : If then Observation 3: If, v then v,
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 Dot Product (Algebraic Definition) Def.: The dot product of to ve vectors v and in here v v, v,..., vn and,,..., n. n is given by v Exercise b : Determine the dot product of each of the folloing pairs of vectors. (i) v 6, 4 3, v (ii) v 6, 4 v 3, v (iii) v 4, 6 3, v (iv) v,, 4 (v) v 5, v, 4 What the Heck is a Dot Product? The dot product is an on. Put another ay, the dot product is a hose input is and hose output is a. Properties of the Dot Product Theorem: Let u, v, and be vectors, and let c be a scalar. Then. u v. u ( v ) 3. c( u v ) 4. v 0 5. v v v and v v 0 if and only if Def.: To vectors u and v are perpendicular or orthogonal if
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 3 Angle Beteen To Vectors Def.: From the geometric definition of the dot product, e can give the angle θ beteen n to nonzero vectors in implicitly explicitly cos(θ) θ Note: Even though the angle beteen the zero vector, 0, and any other vector is not defined by this definition [can you explain hy?], it is convenient to extend the definition of orthogonality to include 0. I.e., 0 is said to be orthogonal to every vector. Exercise : Let u 3 i j and v 4 i 6 j be vectors in. Sketch the to vectors and determine the angle beteen them, rounded to the nearest tenth of a degree. u v u v Thus, θ Exercise 3: Verify the Hint given in Exercise a(iv). v (,) (,4 ) Also, v and Thus, cos(θ) and θ
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 4 Direction Cosines For a nonzero vector in the plane, e measure direction in terms of the angle counterclockise is the positive direction from the positive branch of the x-axis to the vector. In three-dimensional space, it is more convenient to measure direction in terms of the angles beteen a vector v v i v j v k and the three standard unit vectors. 3 z Def.: The direction angles of v are Def.: The direction cosines of v are cos(α) y x cos(β) cos(γ) Exercise 4: Derive the formula for cos(β). Note that β is the angle beteen and. So, cos(β) Exercise 5 (Section.3 #30): Determine the direction cosines and direction angles of v 5 i 3 j k. Note that v Then cos(α) 5 35 α cos 5 35 cos(β) β cos(γ) γ
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 5 Projections and Vector Components We ill no consider a vector u orthogonally projected onto a vector v. We rite this projection vector as proj v u. Because is a scalar multiple of v, as the diagram belo shos, e can rite ( * ) If k > 0, then e have the folloing diagram. cos(θ) ( ** ) By ( * ), By ( ** ), Thus, Finally, then proj v u Note: If k < 0, it can be shon that proj v u is given by the same formula.
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 6 is called the is called the Moreover, u Note: proj v u Exercise 6: Let u 3 i j and v 4 i 6 j. Sketch and compute proj v u (also knon as ) and the vector component of u orthogonal to v (also knon as.) proj v u u - 3 i j -
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 7 Exercise 7: A forty-ton (including load) truck is parked on a road ith a 9% grade. Assume that the only force to overcome is that due to gravity. Compute the force required to keep the truck from rolling don the hill. The force, F, due to gravity pulls the truck don the hill and against the hill. These to component forces, and, are orthogonal to each other. Let v the unit vector along the road cos( ) i sin( ) j, here θ v indicates the force indicates the force that the
Bob Bron, CCBC Dundalk Math 53 Calculus 3, Chapter Section 3 8 Work Def.: The ork, W, done by a constant force F acting along the line of motion of an object is given by W P Q Def.: The ork, W, done by a constant force F not necessarily directed along the line of motion of an object is given by W P Q Def.: The ork, W, done by a constant force F as its point of application moves along the vector PQ is given by either of the folloing: (projection form) (dot product form) W W