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# Calculus w MATH 242

FSU

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This 6 page Class Notes was uploaded by Lisa Wisoky on Monday October 12, 2015. The Class Notes belongs to MATH 242 at Fayetteville State University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/221584/math-242-fayetteville-state-university in Mathematics (M) at Fayetteville State University.

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Date Created: 10/12/15

MATH 242 A LECTURE ON EQUATIONS OF LINES AND PLANES Threedimensional Cartesian coordinate system Let s rst get some basic notation out of the way The 3D coordinate system is often denoted by 1R3 Likewise the 2D coordinate system is often denoted by and the 1D coordinate system is denoted by IR Also as you might have guessed then a general 7 dimensional coordinate system is often denoted by IE5 Next let s take a quick look at the basic coordinate system The threedimensional Cartesian coordinate is de ned by three axes at right angles to each other forming athree dimensional space The three axes are labeled x sometimes called abscissa y ordinate and z applicate The point of intersection Where the axes meet is called the origin which is normally labeled 0 Srx 02 i quot Q my 0 zeaxt 5 P1 Vt 1 It yeaxt 5 EQUATIONS OF LINES 1n the plane to draw alme we need apamt and aslape The slope gives the dxrecmm of the hhe In general duecuon ls gtveh by avecmr so alme ls detehmmedby apoan on the hhe and a vectorpamllel to the hhe Let Jbe avectorparallel to the hhe L and Pxyz be any pomt on the hhe L Let and bre the position vectors of Pand Pnrespecuvely eh ve 4m when seapehewemeza X Y By mangle law forvecwr addition we have oa rum Th 15 vector equation ofa me Smce a and v arepamllel vectors so wehave a e 7 Vector equation takes the form as a b c abc are known as direction numbas o e me and are dented by m and V or m and n Wehave xnynzntabcxyz ltxtymzngtltmtrbtrcgt ltwgt xn tayn tbzn tc x y z W39hlch are the paramemc equations othe lme L Constder xntax yn tby zn tcz xix 421 kn X7 Xn b 272nm j 3 These are called symmetric equations of the line L 1 Find vector equation ofa line that passes through the point 5 l 3 and is parallel to the vectorl39 4 2k Solution Vector equation of a line is r r0 tv Here a 513 and E 139 4 j 2k 14 2 We have gg 513t14 2 513 ltt4t 2tgt 5 11 413 2 5til4tj3 2tk Practice Problems Find vector equation ofa line that passes through the point l03 and is parallel to the vector 2139 4 5k 1 2 Find vector equation ofa line that passes through the point 24 10 and is parallel to the vector lt31 8gt Find parametric equations for the line that passes through the point 5 l 3 and is parallel to the vectorz39 4 2k Solution Vector equation of a line is r r0 tv Here a 513 and E 139 4 j 2k 14 2 We have V r0 tv 513t14 2 513 ltt4t 2tgt x yz 5 11 413 2 x5t yl4t z3 2t are the parametric equations of the line Find equation of a line in parametric and symmetric form that passes through the points A243 and B3ll Solution Here direction numbers are lt l 54 gt a lb 5c 4 Eqs of the line in parametric form are x2t y 4 5t z 34t Eqs of the line in symmetric form are x 2 y 4 z 3 1 5 4 4 x2t y1 t z43t Here a2 b 1 c3 Eqs of the line in symmetric form are x 0y 0z 0 TT Or 2125 2 1 3 Parametric forms are x 2t y t z 3t 5 Equation of the plane is x 3y z 5 Normal vector the plane is n 131 Since the required line is parallel to the normal to the plane therefore direction numbers of both vectors are the same Equation ofthe line through 10 6 is x 1 y 0 z 6 1 3 1 EQUATIONS OF A PLANE A plane in 3space is not uniquely determined by a parallel vector but is uniquely determined by a normal ie perpendicular vector together with a point in the plane Vector Equation of the Plane Let Px y 2 be an arbitrary point in the plane and is a normal vector to the plane Let 7 and 70 be the position vectors of P and P0 respectively We have n n The normal vector a is orthogonal to every vector m the gweh plane In pamemar a is orthogonal to r 0 we have A may 22772 0 a 397 Thxs xs eaned vector equamh of the plane The Scalar Equation ofa Plane Consider a at 7 0 wa r xnynzn aha Then the above equahon becomes as 415 pamyrynaezn 0 X Xh b 02 5 zizh Thxs is the scalar equamh ofaplane passmg through the pomt gamma th the normal veemh a a 55

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