Methods of Adding Vectors Geometrically MCV4U: Calculus & - PDF document

g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Methods of Adding Vectors Geometrically MCV4U: Calculus & Vectors Recall that two vectors are equivalent if they have the same magnitude and direction. This means that vectors can change their positions and remain equivalent, as long as they maintain their magnitudes and directions. Adding and Subtracting Vectors This makes it possible for us to construct diagrams that represent vector addition or subtraction of two or more J. Garvin vectors. J. Garvin — Adding and Subtracting Vectors Slide 1/21 Slide 2/21 g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Methods of Adding Vectors Geometrically Methods of Adding Vectors Geometrically Triangle Method of Vector Addition Example Given two vectors, � AB and � a and � a + � BC , arranged head to tail as Given vectors � b , draw � b . shown below, the resultant � AC is the sum of � AB + � BC . Using the triangle method of vector addition, J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 3/21 Slide 4/21 g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Methods of Adding Vectors Geometrically Methods of Adding Vectors Geometrically Example Parallelogram Method of Vector Addition Given two vectors, � AB and � a and � a + � Given vectors � b , draw � b . AD , arranged tail-to-tail as shown, let � BC = � AD and � DC = � AB . The resultant � AC is the sum of � AB + � BC or � AD + � DC . Using the parallelogram method of vector addition, J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 5/21 Slide 6/21

g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Methods of Subtracting Vectors Geometrically Methods of Subtracting Vectors Geometrically Tail-to-Tail Method of Vector Subtraction Example Given two vectors, � AB and � a and � a − � AC , arranged tail-to-tail as Given vectors � b , draw � b . shown, the resultant � BC is the difference of � AC − � AB . Using the tail-to-tail method of vector subtraction, J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 7/21 Slide 8/21 g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Methods of Subtracting Vectors Geometrically Adding and Subtracting Vectors Alternatively, a vector may be subtracted from another using Example its opposite vector. Using the following diagram, express � AB + � BC as a single Opposite Vector Method of Vector Subtraction vector. Given two vectors, � AB and � AC , arranged tail to tail as shown, let � CD = − � AB = � BA . The resultant � AD is the difference of � AC − � AB . AB + � � BC = � AC J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 9/21 Slide 10/21 g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Adding and Subtracting Vectors Adding and Subtracting Vectors Example Example Using the following diagram, express � DB − � Using the following diagram, express ( � BC + � CD ) + � CB as a single DA as a vector. single vector. DB − � � CB = � DB + � BC = � ( � BC + � CD ) + � DA = � BD + � DA = � DC BA J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 11/21 Slide 12/21

g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Adding and Subtracting Vectors Adding and Subtracting Vectors Example The last example illustrates the associative property of vector addition. Using the following diagram, express � BC + ( � CD + � DA ) as a Properties of Vector Addition and Subtraction single vector. Given vectors � u , � v and � w : • ( � u + � v ) + � w = � u + ( � v + � w ) (associative property) • � u + � v = � v + � u (commutative property) v + � • � 0 = � v (identity property) The zero vector , � 0, has a magnitude of zero and arbitrary direction. Thus, adding a vector to the zero vector results in the original vector. BC + ( � � CD + � DA ) = � BC + � CA = � BA J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 13/21 Slide 14/21 g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Adding and Subtracting Vectors Adding and Subtracting Vectors Example Example Using the following diagram, let � x and � Using the following diagram, let � x and � AB = � BC = � y . AB = � BC = � y . Express � Express � EF in terms of � x and � y . BG in terms of � x and � y . EF = � � CB = − � BG = � � BC + � CG = � BC + � BA = � BC − � BC = − � y AB = � y − � x J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 15/21 Slide 16/21 g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Adding and Subtracting Vectors Adding and Subtracting Vectors Example Example A ship travels 150 km due east of port, then assumes a Using the following diagram, let � x and � AB = � BC = � y . Express � bearing of N50 ◦ E for 100 km. Use trigonometry to determine AD in terms of � x and � y . the displacement of the ship, and its direction. Use the following diagram. AD = � � AB + � BC + � CD = � AB + � BC + � BG = � x + � y + � y − � x = 2 � y J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 17/21 Slide 18/21

g e o m e t r i c v e c t o r s g e o m e t r i c v e c t o r s Adding and Subtracting Vectors Adding and Subtracting Vectors The displacement is | � r | , where r is the resultant vector. Use The direction can be found if we know the measure of ∠ V . the cosine law. Use the sine law. � sin V = sin R u | 2 + | � v | 2 − 2 | � | � r | = | � u || � v | cos R | � v | | � r | � 150 2 + 100 2 − 2 · 150 · 100 cos 140 ◦ = � 100 · sin 140 ◦ � ∠ V ≈ sin − 1 ≈ 235 . 5 km 235 . 5 ≈ 16 ◦ The displacement is approximately 235.5 km, at a bearing of approximately N74 ◦ E. J. Garvin — Adding and Subtracting Vectors J. Garvin — Adding and Subtracting Vectors Slide 19/21 Slide 20/21 g e o m e t r i c v e c t o r s Questions? J. Garvin — Adding and Subtracting Vectors Slide 21/21