18 02A Topic 17 Vectors dot product Read TB 17 3 18 1 18 2 Vectors Two views First the geometric and then the analytic Geometric view Vector length and direction Discuss scaling scalars same vector Length denoted A also called magnitude or norm Addition head to tail Subtraction either tail to tail or A B T B T B oo7 A A B oooo A ooA B oo o oo B Analytic or algebraic view Place the tail of A at the origin the coordinates of the head determine A A ha1 a2 i a1 i a2 j O a1 a2 A a j 2 a1 i Q P Q P O You ve seen the vectors i and j in physics They have coordinates i h1 0i j h0 1i j i Notation 1 a1 a2 indicate as point in the plane 2 ha1 a2 i indicates the vector from the origin to the point a1 a2 Of course this vector can be translated anywhere and ha1 a2 i a1 i a2 j 3 P OP is the vector from the origin to P 4 In print we will often drop the arrow and just use the bold face to indicate a vector i e P P Discuss scaling scalars p Length A a21 a22 Addition a1 i a2 j b1 i b2 j a1 b1 i a2 b2 j ha1 a2 i hb1 b2 i ha1 b1 a2 b2 i Discuss PQ Q P geom and anal continued 1 18 02A topic 17 2 Dot product scalar product Geometric definition algebraic view A B A B cos A B a1 b1 a2 b2 Hard to get geometrically proof Law of cosines won t do in class T A A A B B B A B 2 A 2 B 2 2 A B cos a21 a22 b21 b22 a1 b1 2 a2 b2 2 2 A B cos a1 b1 a2 b2 A B cos QED Algebraic law A B C A B A C Follows from the algebraic view of dot product Unit vectors Special vectors i and j Note i i 1 j j and i j 0 Unit vector u u 1 b Often indicate by u A u A cos A A A 2 A B A B 0 xx xx x Axxxx x x x xx b u b A cos component of A indirection of u Components or projection b is A u b The component of A in the direction of u For a non unit vector the component of A in the direction of B is the component of B b B A in the direction of u Trig identity cos cos cos sin sin Unit vectors u cos i sin j v cos i sin j G cos sin Angle between them is oo7 cos sin Geometric u v u v cos cos cos ooooo oo Analytic u v u1 v1 u2 v2 cos cos sin sin Example P 5 0 Q 1 3 PQ 6i 3j h6 3i R Example Show PQ QR QP 0 P continued oo7 Q ooooo oo 18 02A topic 17 3 Example Find 2 unit vectors parallel to v 3i 4j v 5 u1 34 i 54 j u2 u1 Example Show the sum of the medians of a triangle 0 Median of AB P 12 A B CP 12 B A C Likewise BQ 21 A C B AR 12 B C A Sum of medians CP BQ AR 0 C R Q oooo B o ooo o P A Example Let A 1 2 B 2 3 and C 2 1 Find the cosine of BAC AB AC Let be the angle cos AB AC yy B yy y y AB h1 1i AC h1 3i yy yy y C A 1 3 2 1 cos 2 10 20 5 Example Velocities are vectors O A river flows at 3mph and a rower rows at 6mph What heading should he use to get straight across a river Need sin 36 6 Answer Head at angle of 6 radians upstream from straigh across Same question with river 2 mph row 2 2 mph sin 2 2 2 4 Same question with river 6 mph row 3 mph sin 63 No such Three dimensions Exactly the same except third coordinate a1 i a2 j a3 k a1 a2 a3 Example Show A 4 3 6 B 2 0 8 C 1 5 0 are the vertices of a right triangle B z Two legs of the triangle are AC h 3 2 6i and AB h 6 3 2i AC AB 18 6 12 0 orthogonal 4 A 44 44 44 44 44 44 44 XXXXX 44 XXXXX XXXXX 4 X tt XXXXXXXXX 444 t t XXXX4X tt 44 XXXXX tt t t C y tt t xt
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