exam1-review.pdf - BASIC FORMULAS 1 Qefinitign If a =(a1...

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Unformatted text preview: BASIC FORMULAS 1. Qefinitign, If a = (a1, a2, a3) and b = (b1, b2, b3) , then the dot product of a and b is the number a -b given by a-b =al-b1+ az-b2 + a3-b3. 2.|a|2=a-a = / a12 +a22 +a32. AM If 9 is the angle between the vectors a and b, then a - b = lallblcosa 6. Corollary. If 9 is the angle between the nonzero vectors a and b, then cos6= [2|]; . 7. Two vectors a and b are orthogonal if and only if a- b = 0. . . __ a - b 8.Scalarpro;ectzon of b onto a : compab — W . . . a- b Vector projection of b onto a : pI'OJab = Fa. a .EXERQSES 33. Show that the vector orthab = b -projab is orthogonal to a. m (orthab) -a = (b-projab) -a =b-a - (projab) -a =b-a- Illa—"gum =b-a — b-a =0. 43. Find the angle between a diagonal of a cube and one of its edges. 59mm The diagonal of the cube that begins at the origin and ends at ( 1, l, 1 ) has vector representation (1, 1, 1) . The angle 6) between this vector and the vector of the edge which also begins at the origin and runs alongx — axis is given by (1,1,1)'(1, 0, 0) = l l |(1,1,1)ll<1,0,0 >l J? ands08=cos_1(—)z 55°. J? cos6= ”L4 HIE QRQ§S ERQDDCT 5. 111m The vector a x b is orthogonal to both a and b. 6. I! hegrem If 9 is the angle between a and__b, then I a x b| =-|a| |b|sina TWTWO nonzero vectors a and b are parallel if and only if a x b = 0. 8. 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