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Unformatted text preview: Math 251 Section 3: Review for Test2
Test2 covers Section10.1 through Section 10.7 (except for 10.6) in calculus. Here is the highlight of each section. Section10.1 The distance between two points; the equation of a sphere. Section10.2 Vector addition, subtraction and scalar multiplication (by geometric as well as algebraic methods); the length of a vector; unit vector. Section10.3 The denition of a · b; the angle between two vectors; projection of a vector onto another vector. Section10.4 The denition of a × b; the direction of a × b ; the length of a × b and the geometric interpretation of |a × b|. Section10.5 The vector equation, parametric equations and symmetric equations of a line; the normal vector of a plane; the equation of a plane; the angle between two planes. Section10.7 Vector equations of space curves; derivatives and tangent vectors; nd the tangent line to a space curve. Here is the list of facts and formulas that you need to memorize. 1. The distance between two points is given by
|P1 P2 | = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 . 2. The equation of a sphere with center (x0 , y0 , z0 ) and radius r is given by
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r2 . 3. a · b = a1 b1 + a2 b2 + a3 b3 = |a||b| cos θ .
1 ijk 4. a × b = a1 a2 a3 b1 b 2 b3 and |a × b| = |a||b| sin θ. (The following facts should be understood instead of just memorized.) 5. The vector equation of a line through P0 (x0 , y0 , z0 ) and parallel to the vector v = a, b, c is given by r = r0 + tv or
x, y, z = x0 , y0 , z0 + t a, b, c . 6. The equation of the plane passing through P0 (x0 , y0 , z0 ) with normal vector v = a, b, c is
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0. 7. The tangent vector of a given the space curve r(t) = x(t), y (t), z (t) is
r (t) = x (t), y (t), z (t) . 2 ...
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