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Dot & Cross Product Dot Product (2.11) • Symbolically: • – NOT, NOT, NOT vector multiplication • Numeric Interpretation #1 • = ∗ + ⋯ + ∗ – Result is a scalar. •=• – Example: 4 0 3 • 2 = 4 ∗ 0 + 3 ∗ 2 + −1 ∗ 5 5 −1 =0+6−5 =1 D.P. Application #1 • Look at • – Numerically: 1 ∗ 1 + ⋯ + ∗ – Look familiar? • Length, without the final square root. • Or…the length squared •= Dot Product, cont. • Numeric Interpretation #2: • = ∗ ∗ cos(θ) – Where θ is the angle made when the two vectors are placed tail-to-tail. – Comes from the Law of Cosines Law of Cosines C γ b a A α β c B c 2 a 2 b2 2ab* cos( ) Sort of like the Pythagorean theorem for any triangle (not just right triangles) Law of Cosines => D.P. Interp #2 1. Let = − 2. Let = − 3. Let = − 4. ● = 2 (by D.P App#1) C γ b a A – This equals b in the drawing above. α β c B 5. Similar steps for w and q 6. 2 = 2 + 2 − 2(γ) (Law of Cosines) 7. ( − )●( − ) = ● + ● − 2 cos γ (using step 4) 8. − ● − = ● + ● − 2 ● (using FOIL and distributive law of D.P., 2.13) 9. ● = cos γ (substitue r.h.s of 8 into l.f.s of 7 and simplify) 10.Q.E.D. Let’s look at the 5two interpretations 10 v 5 Let 0 w 3 and 0 Let’s draw a picture: v w θ w θ is ≈ 60 degrees Example, continued 10 v 5 0 5 w 3 0 Theta is ≈ 60.0 degrees. Let’s hypothesize that interpretation#2 and interpretation#1 are both correct, but let’s compare the numbers for this test case…just to be sure. Interpretation#2:v w v * w * cos( ) v 102 52 02 100 25 125 11.18 w 52 (3) 2 02 25 9 31 5.57 v w v * w * cos 11.18 * 5.57 * cos(60) Interpretation#1: v w 10* 5 5 * (3) 0 * 0 50 15 35 31.14 (Note: We guessed on theta, and rounded off the lengths. Otherwise they would be identical) Application of D.P #2 (Calculation of θ) • We can come up with an exact value for θ, given any two vectors using a little algebra and our two definitions of dot product. v w v * w * cos( ) v * w * cos( ) v w v *w v *w v w cos( ) v *w v w 1 cos ( ) cos 1 (cos( )) v *w v w 1 cos ( ) v *w Application of Dot Product #3 θ is the angle between v and w. In each of these cases, think of what cos(θ) would be… w cos(120)=-0.5 θ≈120 w Acute θ θ θ≈45 cos(45)=0.707 v v θ v θ≈120 cos(120)=-0.5 w w Obtuse θ≈180 cos(180)=-1 θ θ v θ≈90 cos(90)=0 w NOTE: We never have to deal with v Right . Application of Dot Product #3 (θ “Categorization”) • We can classify what type of angle is made by two vectors by looking at the sign of the dot product. • Acute: • Obtuse: • Right: v w 0 v w 0 vw 0 If v and w are both unit-length (normalized), we can make some more observations: 1 vˆ w 1 ALWAYS! vˆ wˆ 1 if they are equal (the d.p is close to 1 if they’re in the same general direction). vˆ wˆ 1 if they are opposite (the d.p is close to -1 if they’re in generally opposite directions). Application #4 (projection) Let v,w be two vectors. Consider a triangle w θ How long is a ? A: *cos(θ) Remember: ● = So… a= ● a *cos(θ) The projection of is (whose length is a) =∗ Application #4, cont. • For any two vectors, and … – The projection of onto is – This can be simplified to: • ●∗ 2 • ●∗ ● ● ∗ Application #4, cont. • It works even if they make an obtuse angle Cross Product (5.11) • Symbolically: v w • Again, NOT, NOT, NOT vector multiplication! • The result is a vector. Cross Product (5.11) A little trickier than dot product Only useful (to us) in 3D Imagine two vectors v,w They aren't parallel (or antiparallel) They lie in a plane The plane has a normal. Call it n. y n Define: w v× w= n v How do we compute it? We can use a visualization to remember... z x Cross Product, Numerically X= Y= Z= x y z Vx Vy Vz Wx Wy Wz x y z Vx Vy Vz Wx Wy Wz x y z Vx Vy Vz Wx Wy Wz v y * wz vz * wy v w vz * wx vx * wz vx * wy v y * wx Another C.P. mnemonic resultx vy * wz vz * wy Increase the subscripts by 1, “wrapping” around from z=>x resulty vz * wx vx * wz Increase the subscripts by 1, “wrapping” around from z=>x resultz vx * wy vy * wx Memorize Me Direction Cross product is anticommutative Meaning: v x w = -(w x v) Determining direction of result: If we compute n = v x w: Use right-hand rule Or, if in left handed space, use left-hand rule w n v Properties Additional properties: ||v x w|| = area of parallelogram ||v x w|| = ||v|| ||w|| sin θ v x v = (0,0,0) y n w v z x