Exam 1 Study Guide with Formulas - The following is a list of formulae and theorems of math 23 the materials and thus no proofs or examples are included

Exam 1 Study Guide with Formulas - The following is a list...

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The following is a list of formulae and theorems of math 23 to help you to memorize the materials and thus no proofs or examples are included. If a formula holds for both 2D and 3D cases and the expressions are similar, we only state the 3D case. (1) Distance formula: given two points P = ( x 1 , x 2 , x 3 ) , Q = ( y 1 , y 2 , y 3 ) R 3 , then the Euclidean distance of p and Q is d ( P, Q ) = | PQ | = ( x 1 - y 1 ) 2 + ( x 2 - y 2 ) 2 + ( x 3 - y 3 ) 2 . (2) Equation of spheres and description of balls: The equation of the sphere with center ( a, b, c ) and radius r is ( x - a ) 2 + ( y - b ) 2 + ( z - c ) 2 = r 2 or x 2 + y 2 + z 2 - 2 ax - 2 by - 2 cz + a 2 + b 2 + c 2 - r 2 = 0 . Please see the later classification of quadric surfaces. The (closed) ball with center P = ( a, b, c ) and radius r is B ( P, r ) = B r ( P ) = { ( x, y, z ) R 3 | ( x - a ) 2 + ( y - b ) 2 + ( z - c ) 2 r 2 } . (3) Properties of vectors: given vectors v = x, y, z and w = a, b, c we have The length of v is | v | = x 2 + y 2 + z 2 . v ± w = x ± a, y ± b, z ± c . If α is a scalar, then αv = αx, αy, αz . v is the zero vector if and only if x = y = z = 0. If v = 0, then the unit vector in the direction of v is 1 | v | v . The standard basis vectors i , j , k have components i = 1 , 0 , 0 , j = 0 , 1 , 0 , k = 0 , 0 , 1 . We can write v as a linear combination v = x i + y j + z k . (4) Products of vectors: given vectors v = x, y, z and w = a, b, c , the dot product of v and w is the scalar v · w = xa + yb + zc. Properties of dot product: • | v | 2 = v · v . v · w = w · v . For scalars α, β and vectors v 1 , v 2 , w we have ( αv 1 + βv 2 ) · w = αv 1 · w + βv 2 · w. Let θ be the angle between v and w . Then v · w = | v || w | cos θ. For nonzero vectors v and w , they are perpendicular if and only if v · w = 0. The scalar projection of w onto v is comp v w = v · w | v | and the vector of w onto v is proj v w = v · w | v | v | v | = v · w | v | 2 v. 1
The cross product of two 3D vectors v = x, y, z and w = a, b, c is the vector v × w = det i , j , k x, y, z a, b, c = ( yc - bz ) i + ( za - xc ) j + ( xb - za ) k . Properties of dot product: v × w = - w × v . For scalars α, β and vectors v 1 , v 2 , w we have ( αv 1 + βv 2 ) × w = αv 1 × w + βv 2 × w. Let θ be the angle between v and w . Then | v × w | = | v || w | sin θ. The vector v × w is perpendicular to both v and w and the triple ( v, w, v × w ) satisfy the right hand rule. Furthermore, | v × w | is the area of the parallelo- gram spanned by v and w . For nonzero vectors v and w , they are parallel if and only if v × w = 0. For vectors u, v, w we have u · ( v × w ) = ( u × v ) · w and u × ( v × w ) = ( u · w ) v - ( u · v ) w. The volume of the parallelepiped spanned by u, v, w is V = | u · ( v × w ) | . (5) Lines: a line L in R 3 is determined by a point Q 0 = ( x 0 , y 0 , z 0 ) on L and a nonzero vector v = a, b, c which is parallel to L . The vector equation is r = r 0 + t v where r 0 = --→ OQ 0 and t is a parameter. The parametric equation is x = x 0 + at y = y 0 + bt z = z 0 + ct and the symmetric equation is x - x 0 a = y - y 0 b = z - z 0 c when abc = 0.

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