contestada

The line (x, y, z) = (1 − 2t, 2t − 1, t), with t as the parameter, is denoted by L. The point Q is (1, 2, 3).

a) Find a point P on L such that the line from P to Q is orthogonal (perpendicular) to L.

Respuesta :

Answer:

P = (-1,1,1)

Step-by-step explanation:

for the line

L (t) = (1 − 2t, 2t − 1, t)

since the vector that is tangent to L is its derivative L'=dL/dt , then

L'= dL/dt (t) =  (−2, 2 , 1)

then the vector PQ (P to Q) is

PQ = (x,y,z) - (1, 2, 3) = ( x-1 , y-2 ,z-3)

since P belongs to L → x= 1-2t . y=2t-1 and z=t ,

PQ =  ( x-1 , y-2 ,z-3) = (-2t , 2t-3 , t-3)

to be ortogonal to L , PQ should be ortogonal to L' , then the dot product between PQ and L should be 0

PQ * L' = 0

(-2t , 2t-3 , t-3)* (−2, 2 , 1) =0

4t + (4t-6) + (t-3) = 0

9t - 9= 0

t = 1

then

P = L(t=1) = ( 1-2*1 , 2*1-1 , 1 ) = (-1,1,1)

P = (-1,1,1)

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