vectoralgebrahw1 - ma2176a1 1. Vector Algebra Let a = 2i 2...

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ma2176a1 Vector Algebra 1. Let , 22 aij =− + GGG k G 8 4 bi j k = +− GG G G , 12 4 3 ci j k = −− G . Find (a) a G , a a G G , ab + GG , + G G , , G G ; (b) and the angle between and a G b G ; (c) the coefficient of the projection of b G on a G ; (d) , , × ca × ( ) c ×× G , ( ) ab c GG G ; (e) the area of the triangle with sides a G and c G ; (f) ; ⋅× (g) the volume of the tetrahedron with ac + G G , G G and b G as adjacent edges. 2. Use vector method to find the volume of the tetrahedron of which the vertices are the points () ( ) ( ) ( ) 1,0,1 , 1, 2,3 , 1, 2, 3 , 3, 2,1 AB C D −−− . 3. Using vector method, show that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram. Further, show that the perimeter of this parallelogram is equal to the sum of the lengths of the diagonals of the original quadrilateral. 4. Let L be the line through and 1 1, 2, 2 P ( ) 2 0,2,5 P . Find the trisection points of the line segment . 12 PP 5. Show that for any vectors and
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