Calculus 3 text 2 study guide, Schemes and Mind Maps of Calculus

Download Calculus 3 text 2 study guide and more Schemes and Mind Maps Calculus in PDF only on Docsity! LEV. 5 - Due today (1a. /la.¢) Review Que @ Class fine Wed. Vest Q- Wed. (Ch (a) / questions Chapter 12 Review Section 12.1 - Distance in 3D: |𝑃1𝑃2| = √(𝑥2 − 𝑥1) 2 + (𝑦2 − 𝑦1) 2 + (𝑧2 − 𝑧1) 2 - Equation of sphere: (𝑥 − 𝑥0) 2 + (𝑦 − 𝑦0) 2 + (𝑧 − 𝑧0) 2 = 𝑎2 Section 12.2 - Given points 𝐴(𝑥1, 𝑦1, 𝑧1) and 𝐵(𝑥2, 𝑦2, 𝑧2) the vector 𝐴𝐵⃗⃗⃗⃗ ⃗ = 〈𝑥2 − 𝑥1, 𝑦2 − 𝑦1, 𝑧2 − 𝑧1〉 - Magnitude/length: |𝒗| = √(𝑣1) 2 + (𝑣2) 2 + (𝑣3) 2 - Direction of 𝒗: 𝒗 |𝒗| (also known as unit vector of v) - Midpoint: 𝑀 = ( 𝑥1+𝑥2 2 , 𝑦1+𝑦2 2 , 𝑧1+𝑧2 2 ) Section 12.3 - Dot product: 𝒖 ∙ 𝒗 = 𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3 - Angle between two vectors 𝒖 and 𝒗: 𝜃 = arccos ( 𝒖∙𝒗 |𝒖||𝒗| ) o If 𝒖 ∙ 𝒗 = 0, then the vectors are orthogonal - Scalar component of 𝒖 in the direction of 𝒗: 𝒖∙𝒗 |𝒗| - Vector projection of u onto v: 𝑝𝑟𝑜𝑗𝒗𝒖 = ( 𝒖∙𝒗 |𝒗| ) 𝒗 |𝒗| Section 12.4 - Cross Product: 𝒂 × 𝒃 = | 𝒊 𝒋 𝒌 𝑎1 𝑎2 𝑎3 𝑏1 𝑏2 𝑏3 | = | 𝑎2 𝑎3 𝑏2 𝑏3 | 𝒊 − | 𝑎1 𝑎3 𝑏1 𝑏3 | 𝒋 + | 𝑎1 𝑎2 𝑏1 𝑏2 | 𝒌 - Important property: 𝒖 × 𝒗 = −(𝒗 × 𝒖) - Two nonzero vectors are parallel if and only if 𝒖 × 𝒗 = 0 - Area of parallelogram: 𝐴 = |𝒖 × 𝒗| - Area of triangle: 𝐴 = 1 2 |𝒖 × 𝒗| - Volume of the parallelepiped: 𝑉 = |(𝒖 × 𝒗) ∙ 𝒘| - If the scalar triple product equals 0, then the vectors are coplanar. Find the distance fan the pt (37h 4) § to line x 245A, Y= 3 #34, 2-544 {lest Pte a) 2 PC4,3,-9 ann PBS 3-41-34 C89 2 SUD % = 2-239 = Bx BOG. (-la-|8)z - (-3- “Calls + (-2- 4)ie -) ans = “$07 - 63 -Ck <_ 4. \t00+3c+38¢ | 97a Ll +449 V4 lo) Find egn of the plane theough fo (5°69 AndC perpend. te he _line P, X= -Sta y= h+54 2-34 VaR nz C1, 5,37 A(x- xe) +Bly-v) tC (2-%e) =? | (x--5)) + 5(y-CO) +3 (2 -Y* X45 +5y +30 +3 2-AT7=O [*t5 t24+&=0 (