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0.2 Scalar Product & Projections St Bk + Readings in App A & Stewart Ch 9 _________________________________________________________________________________________________________________________________. What & Why? --------------------------
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0.2 Scalar Product & ProjectionsSt Bk + Readings in App A & Stewart Ch 9_________________________________________________________________________________________________________________________________ What & Why? -------------------------- Scalar or Dot Product is a single number that holds information about the angle between 2 vectors. • We can use it to find that angle or to calculate lengths; • And to test for perpendicularity – very important! • And to calculate Projections & Components. These tell us how far a given vector extends in some direction of interest!
Defn of Dot (or Scalar) Product: (a1, a2) . (b1,b2) = a1b1 + a2b2 NB eg ( 3, -1) . ( 2, 4) = 6 - 4 = 2. The dot product of a vector with itself gives the sq of its length: (a, b) . (a. b) = a2 + b2 . ie u . u = |u|2 NB Know the Rules for Dot Product: Study Book p12. Defn of the angle between two vectors: it is the smallestnon-negative (ie unsigned) angle. Using the Cosine Rule: Th 2, App A, p 166, proves u . v = |u| |v | cos tNB! Hence dot product can be used to find angle t: cos t = u.v NB! | u| |v |
In u . v = |u| |v| cos t, the lengths |u| & |v| are always + ve. Hence the SIGN of u . v is determined by the factor cos t . Now for acute angles, cos t is +ve; for obtuse angles, cos t is -ve. Hence the sign of their dot product of tells us about the size of the angle between two vectors:t they “pull together” if and only if their dot product is +ve. they “pull apart” if and only if their dot product is -ve. t
Tests for Parallel & Perpendicular vectors: Parallel vectors have angle 0 or between them: They are scalar multiples of each other (App A, Th3, p 167). Example: (3, -1) & (-6, 2) are parallel: see that (-6, 2) = - 2 (3, -1) . Orthogonal (ie perpendicular) vectors make a right angle. Substituting cos t = 0 for t = /2 into cos t = u . v / |u||v| gives the dot product test for non-zero vectors u & v: u & v are perpendicular if & only if u . v = 0.
Scalar Component of u in the direction of v: u tv this length ??? How far does u project in the direction of v? Using trig, this scalar component is |u| cos t NB But since u . v = |u| |v| cos t another way to calculate scalar component isu .v / |v| . Eg the scalar projection of (3,1) on ( 3, 4) is (9+4) / 5 . Scalar projection/component is a “signed” distance because cos t is +ve if t is acute, - ve if t is obtuse.
Vector Projection of u on v: To express the scalar component or projection as a vector quantity, we point it in the direction of unit vector v |v| Projv u = u . v v = u . v v | v | | v | | v |2 We can then decompose (or resolve) u into the sum of two orthogonal vectors:u u-p p One is the projection p , which can be found by this formula. The other is then simply u - p , found by subtraction.
Homework: Read Study Book Section 0.2 • Without a calculator, give exact values for the sine, cos & tan of 0 , /2 , , /6 , /4 , /3 , 3 /4 , - /4 . The 30/60/90 degree & 45/45/90 triangles are a big help: 1 sqrt2 2 2 sqrt 3 1 1 1 • Appendix A Problems 3.2: Master 1-16, 21-26, 39. Write full solutions to Q 2, 9, 10, 12, 14, 16, 34, 39. • Study Book Th 1 p 12 : try to prove the rules.
Objectives: Be able to • use dot product to find angles • to determine the relative direction of 2 vectors • spot parallel vectors - as multiples of each other • use dot product to test for perpendicularity • prove & use the rules for dot product • find the scalar component of one vector on another • find the vector projection of one on another • decompose a vector into the sum of two that are mutually orthogonal
