help with part iii and iv please A computer-controlled machine can be programmed
ID: 1859284 • Letter: H
Question
help with part iii and iv please
A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill holes by entering the equation of the line of the hole. A 20 cm times 30cm times 30 cm cuboid is to be cut and drilled. The cuboid is positioned relative to x-,y- and z-axes as shown in Fig. 8.1 First, a plane cut is made to remove the corner at E. The cut goes through the points P. Q and R, which are the midpoints of the sides ED, HA and EF respectively. Write down the coordinates of P, Q and R. Show that the vector (2 3 2) is perpendicular to the plane through P. Q and R. Hence find the cartesian equation of this plane. A hole is then drilled perpendicular to triangle PQR, as shown in Fig. 8.2. The hole passes through the triangle at the point T which divides the line PS in the ratio 2:1, where S is the midpoint of QR. Write down the coordinates of S, and show that the point T has coordinates (-5, 16 2/3, 25). [4J Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C.Explanation / Answer
3) Coordinates of Q = midpoint of AE = (0,(20+20)/2,30/2) = (0,20,15)
Coordinates of S = midpoint of QR = (-30/2,40/2,45/2) = (-15,20,45/2)
Coordinates of P = midpoint of ED = (0,10,30)
Coordinates of T = (2S+1P)/(2+1) = (-30/3, 50/3, 25) = (-5,16*2/3, 25)
4) DR's of QF and PF on the plane = (-30,0,0) and (-30,10,0)
DR's of the normal to the plane = (-30,0,0) X (-30,10,0) = (0,0,-300)
Equation of line passing through T is
r = a + tb
r = (-5i + 50j/3 + 25k) + t(-300k)
Coordinates of C = (-30,0,0)
putting above doesnt satisfy the equation, so it doesnt passes through C
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