261-S12-sg1

261-S12-sg1 - MA 261 Spring 2012 Study Guide 1 1 Vectors in...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA 261 - Spring 2012 Study Guide # 1 1. Vectors in R 2 and R 3 (a) ⃗ v = ⟨ a, b, c ⟩ = a ⃗ i + b ⃗ j + c ⃗ k ; vector addition and subtraction geometrically using paral- lelograms spanned by ⃗ u and ⃗ v ; length or magnitude of ⃗ v = ⟨ a, b, c ⟩ , | ⃗ v | = √ a 2 + b 2 + c 2 ; directed vector from P ( x , y , z ) to P 1 ( x 1 , y 1 , z 1 ) given by ⃗ v = P P 1 = P 1 − P = ⟨ x 1 − x , y 1 − y , z 1 − z ⟩ . (b) Dot (or inner) product of ⃗ a = ⟨ a 1 , a 2 , a 3 ⟩ and ⃗ b = ⟨ b 1 , b 2 , b 3 ⟩ : ⃗ a · ⃗ b = a 1 b 1 + a 2 b 2 + a 3 b 3 ; properties of dot product; useful identity: ⃗ a · ⃗ a = | ⃗ a | 2 ; angle between two vectors ⃗ a and ⃗ b : cos θ = ⃗ a · ⃗ b | ⃗ a || ⃗ b | ; ⃗ a ⊥ ⃗ b if and only if ⃗ a · ⃗ b = 0; the vector in R 2 with length r with angle θ is ⃗ v = ⟨ r cos θ, r sin θ ⟩ : x y θ r (c) Projection of ⃗ b along ⃗ a : proj ⃗ a ⃗ b = { ⃗ a · ⃗ b | ⃗ a | } ⃗ a | ⃗ a | ; Work = ⃗ F · ⃗ D . b proj a proj a b b a a b (d) Cross product (only for vectors in R 3 ): ⃗ a × ⃗ b = ⃗ i ⃗ j ⃗ k a 1 a 2 a 3 b 1 b 2 b 3 = a 2 a 3 b 2 b 3 ⃗ i − a 1 a 3 b 1 b 3 ⃗ j + a 1 a 2 b 1 b 2 ⃗ k properties of cross products; ⃗ a × ⃗ b is perpendicular (orthogonal or normal) to both ⃗ a and ⃗ b ; area of parallelogram spanned by ⃗ a and ⃗ b is A = | ⃗ a × ⃗ b | : b a the area of the triangle spanned is A = 1 2 | ⃗ a × ⃗ b | : b a Volume of the parallelopiped spanned by ⃗ a , ⃗ b ,⃗ c is...
View Full Document

This note was uploaded on 03/26/2012 for the course STAT 350 taught by Professor Staff during the Spring '08 term at Purdue.

Page1 / 5

261-S12-sg1 - MA 261 Spring 2012 Study Guide 1 1 Vectors in...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online