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### 37 Cards in this Set

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 DISTANCE FORMULA |p1, p2| √(x2 - x1)² + (y2 - y1)² + (z2 - z1)² 2 minus 1 Equation of a Sphere with center (h,k,l) and radius r (x - h)² + (y - k)² + (z - l)² = r² vector a quantity that has both magnitude and direction vector components Definition of vector addition If a and v are vectors positioned so the initial point of v is at the terminal point of a then the sum u + v is the vector from the initial point of u to the terminal point of v. PQ + QR = PR QS - PS = QP Properties of vectors a + b = b + a a + (b + c) = (a + b) + c a + 0 = a a + -a = 0 scalar-c(a + b) = ca + cb scalars(c + d)a = ca + da scalars(cd)a = c(da) 1a = a standard basis vectors i <1,0,0> j <0,1,0> k <0,0,1> unit vectors a vector whose length is 1 (standard basis vectors) resultant force the vector sum of several forces acting on an object position vector the representation of a vector from the origin. Given 2 points a = Magnitude or length of a vector |a| = √a1² + a2² + a3² Adding vectors algebraically a + b = dot product a ∙ b = (a1)(b1) + (a2)(b2) + (a3)(b3) Properties of the dot product a ∙ a = |a|² a ∙ b = b ∙ a a ∙ (b + c) = a ∙ b + a ∙ c (ca) ∙ b = c(a ∙ b) = a ∙ (cb) 0 ∙ a = 0 Theorem 3 If Θ is the angle between vectors a and b a ∙ b = |a| |b| cos Θ cos^-1 Θ = a ∙ b ∕ |a| |b| Scalar projection of b onto a comp(a) b = a ∙ b / |a| vector projection of b onto a proj a b = (a ∙ b / |a|²) ( a) Work/Force/Distance Equation W = F ∙ D if no angle or w = |F||D| cos Θ Theorem 4 If Θ is the angle between a and b then |a x b| = |a| |b| sin Θ or sin^-1 Θ = |a x b| / |a| |b| x²/a² + y²/b² + z²/c² = 1 Ellipsoid (3)2nd order terms and a constant) z²/c² = x²/a² + y²/b² Cone (3)2nd order variables No constants. If put in form x²/a² + y²/b² - z²/b² = 0 - Opens towards axis of negative variable. z/c = x²/a² + y²/b² Elliptic paraboloid (2)2nd order terms that have same sign. (1)1st order term. No constants. Opens along 1st order term axis. 2nd order terms determines + or -. z/c = x²/a² - y²/b² Hyperbolic Paraboloid (2)2nd order terms that have opposite signs. (1)1st order term. No constants. Neg 2nd order term = head of horse. 1st order term = top of saddle. x²/a² + y²/b² - z²/c² = 1 Hyperboloid of one sheet (3)2nd order terms (1) constant (1) negative term - opens along this term -x²/a² - y²/b² + z²/c² = 1 Hyperboloid of two sheets (3) 2nd order terms (1) constant (1) positive term - opens along this term x² + y² = 1 Cylinder (2) variables extruded along missing variable vector equation of a line r = rօ + t(v) r = (x,y,z) + t parametric equations x = xօ + at y = yօ + bt z = zօ + ct symmetric equations (x - xօ)/a = (y - yօ)/b = (z - zօ)/c vector equation of a plane n ∙ r = n ∙ rօ or n ∙(r - rօ) = 0 scalar equation of a plane a(x - xօ) + b(y - yօ) + c(z - zօ) = 0 normal vector vector orthogonal to a plane cross product linear equation of a plane ax + by + cz + d = 0 parallel planes normal vectors are parallel Distance from point to a plane |ax₁ + by₁ + cz₁ + d|/√a² + b² + c² Area of a parallelogram |a x b| volume of a parallelepiped |a ∙ (b x c)|