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Question

A line passing through the point A with position vector a=4^i+2^j+2^k is parallel to the vector b=2^i+3^j+6^k. Find the length of the perpendicular drawn on this line from a point P with vector r1=^i+2^j+3^k.

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Solution

Given position vector
a=4^i+2^j+2^k
parallel vector
b=2^i+3^j+6^k
eq of line
r=4^i+2^j+2^k+λ(2^i+3^j+6^k)
Point P(1,2,3)
let a line from point P(1,2,3) meets the vector at point Q(x,y,z) on given line perpendicularly
cartesian form of given line
x42=y23=z26=λ
x=2λ+4
y=3λ+2
z=6λ+2
normal vector of PQ

a2=2λ+41=2λ+3

b2=3λ+22=3λ

c2=6λ+23=6λ1

line PQ is perpendicular to given line so
aa2+bb2+cc2=0

2(2λ+3)+3(3λ)+6(6λ1)=0

4λ+6+9λ+36λ6=0

49λ=0

λ=0

x=4
y=2
z=2
distance from point P

PQ=(41)2+(22)2+(23)2

PQ=32+(1)2

PQ=10



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