Can anyone help me with this vector question?

Question

A pipeline is to be fitted beneath a road as represented in the image. All lengths are in meters.

A) Calculate the length of the pipe AB. Use diagrams to support your answer where required.
B) The section BC is to be drilled in the direction 3i + 4j + k. Find the angle between the sections AB and BC.
C) The pipe ends at point C (a,b,0). Give a vector equation of the line BC. Hence find a and b.

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Answer 1 · 2018-03-24T14:36:15

See below.

Given

#A = (0,-20,0)#
#B = (20,0,-10)#
#C = (a,b,0)#

we have

#bar(AB) = norm(A-B) = sqrt(20^2+20^2+10^2)#

Segment #s_1 = A+lambda (B-A)# with #lambda in [0,1]# has the direction of #B-A = (20,20,-10)#

now with #vec n = (3,4,1) = mu (C-B)# we have

# << B-A, vec n >> = norm(B-A) norm(vec n) cos phi# and then

#cos phi = ( << B-A, vec n >> )/( norm(B-A) norm(vec n)) = (20 xx 3+20 xx 4-10 xx 1)/(sqrt(20^2+20^2+10^2)sqrt(3^2+4^2+1))#

and finally

#C = B + rho vec n# or

#((a),(b),(0)) = ((20),(0),(-10)) + rho((3),(4),(1))#

and solving we get

#rho = 10, b = 40, a = 50#