Vectors, Lines and Planes
Vectors, Lines and Planes Math 2419
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This 6 page Class Notes was uploaded by Saul Cervantes on Tuesday September 29, 2015. The Class Notes belongs to Math 2419 at University of Texas at Dallas taught by Anotoly Ezlydon in Fall 2015. Since its upload, it has received 23 views. For similar materials see Accelerated Calculus II in Mathematics (M) at University of Texas at Dallas.
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Date Created: 09/29/15
VECTORS Vector A distance between two points along with a direction If AX y and BXl y1 then vector AB B A ltXl X yl ygt Zero vector lt0 0gt dAB distance between A and B IIABII Note The notation of the two vertical bars is equivalent to the concept of norm or length of a vector In this case IIABII means the length of vector AB Formula x2 x12 yZ yl Ifu ltX ygt andv ltXl y1gt then u v ltX X1 y ylgt Properties of Vector operations 0 u v v u o uvwuvw o u zero vector u 0 u u 0 o cdu cdu where c and d are both constants c cuvcucv o cducudu o 1u u 0 cu Id u Special Vectors o ilt100gt o jlt010gt o klt001gt For any 0t ltcos0t sindgt is equal to a unit vector A sphere or circle is a set of all points equidistant from a given point where the given point is the center and the distance between all the other points and the center is the radius The standard equation of a sphere is R2 Xa2 yb 2 zc2 where the center is at a b c The standard equation for a cylinder is X2 y2 1 Two vectors are parallel if ltul u2 u3gt ltcvl CV2 CV3gt for some constant c Dot Product Given u ltXl ylgt and V ltX2 y2gt then uV Xlyl X2y2 Example u lt1 2 3gt and V lt9 8 7gt uV 1928379 16 21 4 Properties of the Dot Product 0 uv vu o cuv cuv ucv Dot Product Formula uv u vcos0t uv IIuII39IIVII Two vectors are caller orthogonal if the angle between them is equal to 90 degrees COSU Two nonzero vectors are perpendicular orthogonal if and only if uv 0 Projections uv IIvIIZ Formula for the projection of u onto v projVu Cross Product A matrix is a rectangular array of numbers If the number of rows equals the number of columns then the matrix is called a square matrix Given vectors u ltul u2 u3gt and v ltvl v2 v3gt then 1 k uZ u3 u1 u3 u1 uZ uxvdetu1 uZ u3 det tdet det k 171 v2 v3 v2 v3 v1 v3 v1 v2 Algebraic Properties of Cross Product 0 u X 12 v X u o uXvwuXvuXw o cu X 12 cu X 12 u X CU where c is a constant 0 u X zero vector O o u X u zero vector 0 Scalar Triple Product u 12 X w u X 12 W Geometric Properties of the Cross Product 0 Ifu X visorthogonaltouandV then u Xvu0andu Xv v0 0 H u X 12 u V sind Area of parallelogram defined by V and u I 7 f 3 If Area V height V u sinq o u X 12 0 if and only ifu is parallel to V Lines and Planes Parametric equation of a line Lt tu A Where u is a vector ltul u2 u3gt Example Write the parametric equation of a line given these points A l 2 l and B 304 uABlt3 l0 24 lgtlt225gt Lt lt2 2 5gtt l 2 l so Xt 2t l yt 2t 2 and zt 5t l To check if two lines intersect set them equal to each other If there is a solution then they do intersect Equation of a Plane in Three Dimensions Given the plane aX X1 by y1 Cz 21 0 then 0 the normal vector n is equal to lta b cgt 0 Point on the line P has coordinates X1 yl 21 0 Two planes are parallel if their normal are parallel 4 In the above example example number 5 since the solution makes no sense the line and the plane are parallel
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