Parallelogram Law of Vector Addition



*Reduced Calculator Challenge!Warm up: How do you add 3 + 4 and get 5?Example 1Describe a scenario that would involve finding u+v given the vectors below? How would we add these vectors?121920010541000-3143259906000Triangle Law of Vector AdditionTo find the sum of two vectors u and v using the triangle law of vector addition, draw the two vectors __________________. The sum u+v, or resultant, is the vector from the __________ of the first to the __________ of the second. From the vectors in Example 1, does it matter if we add u+v or if we add v+u? Sketch a diagram below.Parallelogram Law of Vector AdditionTo find the sum of two vectors using the parallelogram law of vector addition, draw the two vectors tail to tail. Complete the parallelogram with these vectors as sides. The sum u+v is the diagonal of the parallelogram from the point where the tails are joined to the point where the heads meet. 42862502730500The angle between two vectors is the angle ≤180° formed when the vectors are placed tail to tail; that is, starting at the same points. Example 2 Two vectors a and b have an angle between them of 60°, and magnitudes of a =3 and b=2. Find a+b. No Calculator.Triangle Inequalityu+v≤u+vWhen does u+v=u+v? Draw two vectors u and v, that would satisfy this scenario. 54578251692400402907510160000How do we subtract vectors? u-v= Connection back to the parallelogram: Where do we see vector u-v in the parallelogram we can draw with u and v?Example 3Given the three vectors a, b and c, sketch i) a+b+c ii) a-b+ciii) b-c-a 518223535560005583555913765c00c554545511430a00a638365541910006040755-2540b00b56978553429000Example 4Find the magnitude and direction of the sum of two vectors u and v if their magnitudes are 5 and 8 respectively and the angle between them is 30°. *Use calculator for one calculationNote - The direction of the resultant u+v is expressed as an angle relative to one of the given vectors. It doesn’t matter which one you choose, provided you state it clearly. Example 5If m and n are unit vectors that make a 45° angle with each other, calculate 4m+7nExample 6 Use the rectangular box above to give a single vector equivalent for each of the 4972050-127000following.a) b) c) HYPERLINK "" Question: What assumptions have we made so far about our operations with vectors?Although they seem so basic or fundamental that we wouldn’t question them, applying simple rules from one number set to another, like the Commutative Property, doesn’t always work. We need to determine what these new basic rules are to ensure that we do simple calculations correctly. PROPERTIES OF VECTOR ADDITIONCommutative Propertya+b=Note: Although vector addition is commutative, not all vector operations are commutative (i.e. cross product – we will deal with this later). Associative Propertya+b+c=Distributive Propertyka+b=Adding Zeroa+0=PROPERTIES OF SCALAR MULTIPLICATIONAssociative Propertymna=Distributive Propertym+na=Example 7Simplify the following using properties of vectors. Explain the property you used at each step. 86x+5y-4(6x-9y)Example 8 Using vector properties, find a+b-2c given a=2i-3j+k, b=i+j+k, and c=2i-3k.50120552984500Example 9Determine RQ+SR+TS+PT as a single vectorChallengeGiven that a=8, b=15, and a+b=20, determine |a-b| ................
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