(a) Show that the points A=(2,1,3), B=(4,4,1), C=(0,2,-2) and D=(-2,-1,0) descri

Question

(a) Show that the points A=(2,1,3), B=(4,4,1), C=(0,2,-2) and D=(-2,-1,0) describe a parallelogram. (b) Find the components of the vector . Hence find a vector equation of the line through A and C (use ? as the line variable). (c) Find the components of the vector . Hence find a vector equation of the line through B and D (use ? as the line variable). (d) If the lines AC and BD intersect at M, find the co-ordinates of the point M. (Hint: Equate the vector equations from (b) and (c) to generate three equations in ? and ?. Solve them simultaneously.) (e) Show that M is the mid-point of AC and of BD.

Explanation / Answer

(a) Show that the points A=(2,1,3), B=(4,4,1), C=(0,2,-2) and D=(-2,-1,0) describe a parallelogram. EASIEST WAY IS TO SHOW THAT THE DIAGNALS BISECT EACH OTHER ..... MID POINT OF AC IS ...........0.5[(2+0) , (1+2) , (3-2) ] = [1 , 1.5 , 0.5] MID POINT OF BD IS............ 0.5[(4-2) , (4-1) , (1+0) ] =[1 , 1.5 , 0.5] BOTH ARE SAME SO DIAGONALS AC AND BD BISECT EACH OTHER SO IT IS A PARALLELOGRAM (b) Find the components of the vector .????? WHICH VECTOR ? Hence find a vector equation of the line through A and C (use ? as the line variable). AC IS VECTOR .....C-A = [0-2 , 2-1 , -2-3 ] = [-2 , 1 , -5 ] EQN. OF LINE AC IS R = [2,1,3] + T[-2 , 1 , -5] WHERE T IS A SCALAR (c) Find the components of the vector .???? WHICH VECTOR ? Hence find a vector equation of the line through B and D (use ? as the line variable). BD IS VECTOR....... D-B = [ -2-4 , -1-4 , 0-1 ] = [ -6 , -5 , -1 ] EQN. OF LINE BD IS R=[4 , 4 , 1 ] + S [ -6 , -5 , -1 ] WHERE S IS A SCALAR (d) If the lines AC and BD intersect at M, find the co-ordinates of the point M. (Hint: Equate the vector equations from (b) and (c) to generate three equations in ? and ?. Solve them simultaneously.) M IS FOUND ABOVE ALREADY UNDER a) MID POIN T M IS ......[1 , 1.5 , 0.5] (e) Show that M is the mid-point of AC and of BD. SHOWN ABOVE UNDER a) (a) Show that the points A=(2,1,3), B=(4,4,1), C=(0,2,-2) and D=(-2,-1,0) describe a parallelogram. EASIEST WAY IS TO SHOW THAT THE DIAGNALS BISECT EACH OTHER ..... MID POINT OF AC IS ...........0.5[(2+0) , (1+2) , (3-2) ] = [1 , 1.5 , 0.5] MID POINT OF BD IS............ 0.5[(4-2) , (4-1) , (1+0) ] =[1 , 1.5 , 0.5] BOTH ARE SAME SO DIAGONALS AC AND BD BISECT EACH OTHER SO IT IS A PARALLELOGRAM (b) Find the components of the vector .????? WHICH VECTOR ? Hence find a vector equation of the line through A and C (use ? as the line variable). AC IS VECTOR .....C-A = [0-2 , 2-1 , -2-3 ] = [-2 , 1 , -5 ] EQN. OF LINE AC IS R = [2,1,3] + T[-2 , 1 , -5] WHERE T IS A SCALAR (c) Find the components of the vector .???? WHICH VECTOR ? Hence find a vector equation of the line through B and D (use ? as the line variable). BD IS VECTOR....... D-B = [ -2-4 , -1-4 , 0-1 ] = [ -6 , -5 , -1 ] EQN. OF LINE BD IS R=[4 , 4 , 1 ] + S [ -6 , -5 , -1 ] WHERE S IS A SCALAR (d) If the lines AC and BD intersect at M, find the co-ordinates of the point M. (Hint: Equate the vector equations from (b) and (c) to generate three equations in ? and ?. Solve them simultaneously.) M IS FOUND ABOVE ALREADY UNDER a) MID POIN T M IS ......[1 , 1.5 , 0.5] (e) Show that M is the mid-point of AC and of BD. SHOWN ABOVE UNDER a)

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