Summary of Math 10C Concepts(3).pdf - SECTIONS 9.1 THROUGH...

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SECTIONS 9.1 THROUGH 9.4 Distance Formula in Three Dimensions: he distance between the points P x , , ) and Q x , , ) is T = ( 1 y 1 z 1 = ( 2 y 2 z 2 : ( x ) 2 x 1 2 + ( y ) 2 y 1 2 + ( z ) 2 z 1 2 Equation of a Sphere: he equation of a sphere with center ( a , , ) and radius r is T b c : ( x ) a 2 + ( y ) b 2 + ( z ) c 2 = r 2 Properties of Vectors: iven the points A x , , ) and B , , ), the vector v from A to B is G = ( 1 y 1 z 1 = ( x 2 y 2 z 2 : y , z AB = v = < x , 2 x 1 2 y 1 2 z 1 > he magnitude ( or length ) of a vector v , , is | v | T = < a b c > : = a 2 + b 2 + c 2 wo vectors u and v are parallel if v u for some ( nonzero ) constant c T = c Dot Product Properties: he dot product of u and v is u u || v | cos θ where θ is the angle between u and v (0 ) T : v = | ≤ θ ≤ π he vectors u and v are orthogonal if and only if u T v = 0 f u , , nd v , , , then u x + y + z I = < x 1 y 1 z 1 > a = < x 2 y 2 z 2 > v = x 1 2 y 1 2 z 1 2 or any vector u , u F u = | u | 2 or two vectors u and v , u F v = v u calar projection of v onto u v v | cos θ ( where θ is the angle between u and v ) S : comp u = | = | u | u v ector projection of v onto u proj v ) u V : u = ( | u | u v u | u | = | u | 2 u v Cross Product Properties: he cross product of u and v is | u || v | sin θ) n , where n is the unit vector perpendicular T : u × v = ( o u and v . t hus , u is a vector perpendicular to both u and v . T × v f u , , nd v , , , then u z z , z x x , x y y I = < x 1 y 1 z 1 > a = < x 2 y 2 z 2 > × v = < y 1 2 y 2 1 1 2 z 2 1 1 2 x 2 1 > Refer to cross product formula in determinant form from lecture on October 8 for those comfortable ( ith determinants .) w f u and v are nonzero vectors , then u if and only if u and v are parallel I × v = 0 or two vectors u and v , u F × v = − v × u or the standard unit vectors i , j , and k i , j , k F : × j = k × k = i × i = j u | rea of parallelogram with sides u and v | × v = A olume of parallelepiped determined by vectors u , v , and w | u v )| V : • ( × w
SECTIONS 9.5, 10.1, 10.2, and 11.1 Equations of Lines: 0 c >

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