MATH1131 Assignment .docx - MATH1131 ASSIGNMENT z5257687 Intersection of Planes We are given two planes in parametric form x1 2 1 3 \u03c0 1 x 2 = \u22124 \u03bb1

# MATH1131 Assignment .docx - MATH1131 ASSIGNMENT z5257687...

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MATH1131 ASSIGNMENT z5257687 Intersection of Planes We are given two planes in parametric form: π 1 : ( x 1 x 2 x 3 ) = ( 2 4 2 ) + λ 1 ( 1 2 1 ) + λ 2 ( 3 4 0 ) π 2 : ( x 1 x 2 x 3 ) = ( 2 4 3 ) + μ 1 ( 1 2 2 ) + μ 2 ( 3 4 1 ) Where x 1 , x 2 , λ 1 , λ 2 1 , μ 2 R Let L be the line of intersection of π 1 and π 2 a. Find vectors n 1 and n 2 that are normals to π 1 and π 2 respectively and explain how you can tell without performing any extra equations that π 1 and π 2 must intersect in a line. To find the normal vector to a plane, take the cross product of its direction vectors. For π 1 : n 1 = ( 1 2 1 ) × ( 3 4 0 ) = ( 4 3 2 ) For π 2 : n 2 = ( 1 2 2 ) × ( 3 4 1 ) = ( 6 5 2 ) b. Find Cartesian equations for π 1 and π 2 . To find the Cartesian equation: n . p = n . x For π 1 : ( 4 3 2 ) . ( 2 4 2 ) = ( 4 3 2 ) . ( x 1 x 2 x 3 ) π 1 : 4 x 1 + 3 x 2 2 x 3 = 0 For π 2 :
( 6 5 2 ) . ( 2 4 3 ) = ( 6 5 2 ) . ( x 1 x 2 x 3 ) π 2 :6 x 1 + 5 x 2 2 x 3 =− 2 c. Method 1: Assign one of x 1 , x 2 or x 3 to be the parameter w and then use your two Cartesian equations for π 1 and π 2 to express the other two variables in terms of w and hence write down a parametric vector form of the line of intersection L . 4 x 1 + 3 x 2 2 x 3 = 0 ( 1 ) 6 x 1 + 5 x 2 2 x 3 =− 2 ( 2 ) Through subtraction of equations: ( 2 ) ( 1 ) :2 x 1 + 2 x 2 =− 2 x 1 + x 2 =− 1 Let x 1 = w : w + x 2 =− 1 x 2 =− 1 w Multiply the first two equations by 3 and 2 respectively to get: ( 1 ) × 3:12 x 1 + 9 x 2 6 x 3 = 0 ( 3 ) ( 2 ) × 2:12 x 1 + 10 x 2 4 x 3