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MCV4U1UNIT 7: APPLICATIONS of VECTORSDateDayTopicHomework17.1 Vectors as ForcesPg. 362 # 2, 3, 5, 6, 8, 10, 11, 13, 14,16- 1727.2 VelocityPg. 369 # 1-837.3 The Dot Product of 2 Geometric VectorsPg. 377 # 1, 2, 5, 6-7 (odd), 8, 9, 11, 1247.4 The Dot Product of Algebraic VectorsPg. 385 #1, 2, 4, 6-8, 10, 1157.5 Scalar and Vector ProjectionsPg. 398 # 1-3, 6-8, 12, 1467.6 The Cross Product of 2 VectorsPg 407 #4, 5, 8, 9, 11Pg 414 #2, 3(b,c,d), 5, 7, 8, 10 77.7 The application of Cross and Dot ProdcutPg 414 #2, 3(b,c,d), 5, 7, 8, 108Chapter ReviewPg. 388 #2, 4, 5, 7, 8, 11, 14 Pg. 418 # 1-7, 9-18, 20, 21, 23, 26, 37 9Chapter TestLearning GoalsIn this unit, we will learn:Vectors as Forces and VelocitiesThe Dot and Cross ProductsApplications of Dot Product and Cross ProductDay1: 7.1 Vectors as ForcesA force causes an object to undergo acceleration. For example, forces push you back in your seat when the car you are in accelerates.The magnitude of force is measured in Newtons (N). At the earth’s surface, gravity causes objects to accelerate at a rate of approximately 9.8 m/s2.Forces are vectors. The single force , the resultant force, that has the same effect as all the forces acting together can be found by vector addition. When an object is in a state of equilibrium (a state of rest or a state of uniform motion), the net force is zero or R=0. For example, steady speed. 414718552768500057467500The equilibrant, , is the opposite force of the resultant force , it is the force that would counterbalance the resultant force. 13335017018000383159019939000Draw an equilibrant for the following system of forces.38150805524500Given three forces on a plane, a state of equilibrium is maintained if a triangle can be formed with the three forces. This can only be done if the triangle inequality holds true that the sum of any two sides must be greater than or equal to the third side.Ex 1: Which of the following sets of forces acting on an object could produce equilibrium?a) 13N, 27N, 14N b) 12N, 26N, 13NEx 2: Find the magnitude and direction of the resultant and equilibrant of a system of forces of 2000N and 1000N acting at an angle of 60o to each other.Algebraic Resultant force:To resolve a vector means taking a single force and decomposing it into two components. A vector can be resolved into its corresponding horizontal and vertical components by creating a right triangle with the given vector. The magnitudes of the vertical and horizontal components can be found using primary trigonometric ratios and a given angle.0444500490918524447500250507520129500Ex 3: In order to keep a 250 kg crate from sliding down a ramp inclined at , the force that acts parallel to the ramp must have a magnitude of at least how many Newtons?Ex 4: An object is being towed by two ropes. The direction of forces of the ropes are and . If the resultant force is 1000N due north, find the magnitude of the tensions of each rope.Ex 5: A box with a force of 100N is hanging from two chains attached to an overhead beam at angles of 50o and 30o to the horizontal. Determine the tensions in the chains.Day 2: 7.2 VelocityVelocity is a vector quantity, as the direction of motion as well as magnitude is important. Speed is the magnitude of velocity.In velocity applications, the resultant (ground speed) is the speed of a plane or boat relative to a person on the ground which includes the effect of wind or current on the air/water speed.A key step to solving a problem is to find an angle in the triangle formed by the vectors whose directions are given. It is helpful to draw small axes at the tail or head of the vectors when drawing diagrams.Ex 1: A plane travels due north at an airspeed of 900 km/h. It encounters a wind blowing at 80 km/h from the west. What is the resultant velocity of the plane?Ex 2: A man can swim 3.5 km/h in still water. Find at what angle to the bank he must head if he wishes to swim directly across a river flowing at a speed of 2 km/h.Ex 3: A plane is steering at an air speed of 525 km/h. The wind is from at 98 km/h. Find the ground speed and course of the plane.Ex 4: A boat heads west of north with a water speed of 3 m/s. Determine its velocity relative to the ground when there is a 2 m/s current from east of north.Day 3: 7.3 The Dot Product of 2 Geometric Vectors The Dot product of two geometric vectors 273113517970500The dot product of two vectors is a scalar (also called the scalar product).92265511049000Properties of the Dot Product:62484060325Two nonzero vectors are perpendicular . if a ⊥ b then a ? b=0Commutative property: a?b= b?a .Associative property with a scalar k: ka?b= a?kb=k( a?b) .Distributive property: a?b+ c= a?b+ a?c Magnitudes property: a?a=|a|2 00Two nonzero vectors are perpendicular . if a ⊥ b then a ? b=0Commutative property: a?b= b?a .Associative property with a scalar k: ka?b= a?kb=k( a?b) .Distributive property: a?b+ c= a?b+ a?c Magnitudes property: a?a=|a|2 Ex1: Calculate if and .Ex2: Calculate the angle between given .Ex3: Expand and simplify:a) b) c) Application of the Dot Product:Work is done when a force acting on an object causes a displacement of an object from one position to another. Work is a scalar quantity measured in joules (J).Work is defined as the dot product: where is the force acting on an object (N), is the displacement caused by the force (m) and is the angle between .Ex5: A 25 kg box is located 8 m up a ramp inclined at an angle of to the horizontal. Determine the work done by the force of gravity as the box slides to the bottom of the ramp.Day 4: 7.4 The Dot Product of Algebraic Vectors Recall: Algebraic vectors in component form in R3.40005257175In R2, if .In R3, if .0In R2, if .In R3, if .Ex 1: Given , find .Using Dot product to find the angle between 2 vectors 213550597155if cosθ=0 ? θ=90° cosθ >0? 0° < θ <90°cosθ<0 ? 90°< θ<180°if cosθ=0 ? θ=90° cosθ >0? 0° < θ <90°cosθ<0 ? 90°< θ<180° 4394208191500Ex2: Determine the angle between and Ex3: Determine the value of k so that are perpendicular (orthogonal).Ex4: A parallelogram has its sides determined by a=2, 3and b=(3, 1) . Determine the angle between the diagonals of the parallelogram formed by these vectors.Ex5: Given vectors , determine the components of a vector perpendicular to each of these vectors.Ex6: Suppose that a force vector given by F=(-2, -6, 3) moves an object from point A(3, -1, 2) to point B(1, 4, 4). Calculate the work done on the object.Day 5: 7.5 Scalar and Vector Projections Projections are formed by dropping a perpendicular from the head of one vector to another vector, or an extension of another vector. (Can be thought of as a shadow)Given two vectors a and b, think of the projection of a on b as the shadow that a casts on b 3359785131445003587753619500The direction of the projection of a on b depends on the angle θ between a and b-16256048260001518285571500790575-381000010261604445000Example1: Find the scalar and vector projections of .4373880328295zyxzyxThe angles that a vector makes with each positive axis are called direction cosines. is the angle makes with the positive x-axis. is the angle makes with the positive y-axis. is the angle makes with the positive z-axis.Example2: Determine the direction angles for . Day 6: 7.6 The Cross Product of 2 Vectors -5143510731500The cross product (vector product) is defined only in R3 since the cross product of two vectors is a vector that is perpendicular to both .The cross product of the vectors is the vector whose direction is perpendicular to , such that form a right-handed system42221156540500234124513335000323859969500The vector a × b is the opposite of b × a and points in the opposite direction.The magnitude of the cross product of two vectors is: where is the angle between , .Algebraically:Given: -1206595250011239510223500Another view of Cross Product:Finding a vector perpendicular to two vectors:If are two non-collinear vectors in R3, then every vector perpendicular to both is in the form .251523519050Let be vectors in a × b= -b ×a (not commutative)a ×b+ c= a × b+ a × c (distributive law)ka×b= a×kb=k( a × b)00Let be vectors in a × b= -b ×a (not commutative)a ×b+ c= a × b+ a × c (distributive law)ka×b= a×kb=k( a × b)Properties of the Cross Product:Examples:Find a vector perpendicular to both (1, 2, 3) and (-1, 0, 4).If and the angle between is , find .Day 7: 7.7 Applications of Cross and Dot Product 38931851816100021672557048500Area of a Parallelogram: Ex1: Determine the area of the parallelogram determined by the vectors p=-1, 5, 6 and q=(2, 3, -1)Ex2: Find the Area of the triangle ABC whose vertices are A(1, 3, 5), B(-2, -3, 4) and C(0, 3, -1).Torque is a vector quantity measured in Newton-metres (N-m) (or in joules (J)). Force causes an object to turn causing an angular rather than a linear displacement. The torque is caused by a force defined as the cross product. or where is the applied force, in Newtons and is the vector acting on the axis of rotation, in meters, is the angle between and .Ex3: A force of 90N is applied to a wrench 15 cm long at to the handle. Determine the torque.Review7.1 – Vectors as ForcesResultant Force: single force used to represent the combined effect of all the forces;F = F1 + F2Equilibrant force: single force that opposes the resultant force -F = -(F1 + F2)Resolution: decomposing a force into its horizontal and vertical componentsIf a + b + c = 0, then a, b and c are in a state of equilibrium7.2 – Velocity Velocity stated relative to a frame of reference ex: the ground, the river bankPlanes and wind with compass directions, boats crossing rivers with currents.Can resolve into horizontal and vertical components, or can use cosine or sine law.Vr = V1 + V27.3 – Dot Product of Two Geometric VectorsGeometric vectors: do not have an associated coordinate systema? b=|a||b|cosθif a ⊥ b then a ? b=0if 0<θ <90° then a?b >0 if 90°<θ <180° then a?b <0Commutative property: a?b= b?aDistributive property: a?b+ c= a?b+ a?cMagnitudes property: a?a=|a|2Associative property with a scalar k: ka?b= a?kb=k( a?b)7.4 – Dot Product of Algebraic Vectorsa?b= a1b1+ a2b2+ a3b37.5 – Scalar and Vector ProjectionsScalar projection of a on b is acosθ= a?b|b| b on a is bcosθ= a?b|a|Vector projection of a on b is a?b|b|2bProjection angles: x axis: Cos α = a / √(a2+b2+c2) y axis: Cos β = b / √(a2+b2+c2) z axis: Cos γ = c / √(a2+b2+c2?)7.6 – Cross Product of Two Vectors (only in R3)Finding a vector that is perpendicular to each of the two given vectorsa × b= (a2b3- a3b2 , a3b1- a1b3 , a1b2- a2b1)a × b= -b ×a a ×b+ c= a × b+ a × c ka×b= a×kb=k( a × b)7.7 – Applications of the Dot Product and Cross Product Work:W = |F||s|cosθ and W = F ? sTorque:T = |r||F|sinθ and T = | r x F| (include vector signs) ................
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