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Addition of vectors (i) Triangle Rule [For vectors with a common point] A B C

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(ii) Parallelogram Rule [for vectors with same initial point] A B C D

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(iii) Extensions follow to three or more vectors p q r p+q+rp+q+r

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Subtraction First we need to understand what is meant by the vector – a – a a a and – a are vectors of the same magnitude, are parallel, but act in opposite senses.

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A few examples b a b a b – a

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p q a)Which vector is represented by p + q ? b)Which vector is represented by p – q ? – q p – q p q p + q p

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A C B CB = CA + AB = - AC + AB = AB – AC

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Position Vectors Relative to a fixed point O [origin] the position of a Point P in space is uniquely determined by OP O P OP is a position vector of a point P. We usually associate p with OP p

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A very Important result! O b a AB = b - a A B

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The Midpoint of AB O B A b a OM = ½( b + a) M

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(a) (b) (c) (d) (e) (f) (g) (h) Vectors Questions Example In the diagram, OA = AP and BQ = 3OB. N is the midpoint of PQ. and A P N Q O B a b Express each of the following vectors in terms of a, b or a and b.

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A P N Q O B a b a) b) c) d) e) f) g) OA = AP and BQ = 3OB h)

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Example ABCDEF is a regular hexagon with, representing the vector m and, representing the vector n. Find the vector representing A B C D F E m n m m+nm+n

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M, N, P and Q are the mid-points of OA, OB, AC and BC. OA = a, OB = b, OC = c (a)Find, in terms of a, b and c expressions for (i) BC (ii)NQ (iii) MP (b)What can you deduce about the quadrilateral MNQP? Example c b a (i)BC = BO + OC = c – b (ii) NQ = NB + BQ = c (ii) MP = MA + AP = c MNPQ is a parallelogram as NQ and MP are equal and parallel. a)

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Example In the diagram, and, OC = CA, OB = BE and BD : DA has ratio 1 : 2. O D B C E A b a a) Express in terms of a and b (i) (ii) (iii) (iv) b) Explain why points C, D and E lie on a straight line.

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Example ABC is a triangle with D the midpoint of BC and E a point on AC such that AE : EC = 2 : 1. Prove that the sum of the vectors,, is parallel to

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