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Question

Find the vector and the Cartesian equations of the line that passes through the points (3,2,5),(3,2,6)

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Solution

Let the line passing through the points P(3,2,5) and Q(3,2,6) be PQ. Since PQ passes through P(3,2,5), its position vector is given by,
a=3^i2^j5^k
The direction ratios of PQ are given by,
(33)=0,(2+2)=0,(6+5)=11
The equation of the vector in the direction of PQ is
b=0^i0^j+11^k=11^k
The equation of PQ in vector form is given by r=a+λb,λR
r=(3^i2^j5^k)+11λ^k
The equation of PQ in Cartesian form is
xx1a=yy1b=zz1c
x30=y+20=z+511

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