Question 1040746: Helloo amazing tutors, can you guys help me out in answering this? Thank youu
The points A, B, C and D have position vectors 3i + 2k, 2i − 2j + 5k, 2j + 7k and −2i + 10j + 7k
respectively.
(i) Use a scalar product to show that BA and BC are perpendicular
(ii) Show that BC and AD are parallel and find the ratio of the length of BC to the length of AD
Found 2 solutions by robertb, Fombitz:
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Let O be the origin (0,0,0).
Then vector OA = <3,0,2>,
vector OB = <2,-2,5>,
vector OC = <0,2,7>, and
vector OD = <-2,10,7>.
(i)
==> BA = OA - OB = <1,2,-3>. Also,
BC = OC - OB = <-2,4,2> = 2<-1,2,1>.
==>BA*BC = 1*-2 + 2*4 + -3*2 = -2+8-6 = 0.
==> Vectors BA and BC are perpendicular.
(ii)
Now AD = OD - OA = <-5,10,5> = 5<-1,2,1>.
Since both BC and AD are non-zero multiples of the same vector <-1,2,1>, these two vectors are parallel.
==>  .
Answer by Fombitz(32388)  (Show Source):
You can put this solution on YOUR website! If the vectors are perpendicular, then the dot product equals zero.
A=(3,0,2)
B=(2,-2,5)
C=(0,2,7)
D=(-2,10,7)
So then,
BA=(2-3,-2-0,5-2)=(-1,-2,3)
BC=(2-0,-2-2,5-7)=(2,-4,-2)
The dot product is,
Complete that to verify that the dot product is zero.
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.
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AD=(3-(-2),0-10,2-7)=(5,-10,-5)
AD=5*(1,-2,1)
Similarly for BC,
BC=(2,-4,2)
BC=2*(1,-2,1)
So they are both multiples of the same vector and the ratios of their lengths is equal to the ratio of the multipliers,
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