Tamil Nadu Board of Secondary EducationHSC Arts Class 11th

# The position vectors of the points P, Q, R, S are ijkijijki^+j^+k^,2i^+5j^,3i^+2j^-3k^, and ijki^-6j^-k^ respectively. Prove that the line PQ and RS are parallel - Mathematics

Sum

The position vectors of the points P, Q, R, S are hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2hat"j" - 3hat"k", and hat"i" - 6hat"j" - hat"k" respectively. Prove that the line PQ and RS are parallel

#### Solution

Given that the position vector of the given points P, Q, R, S are

vec"OP" = hat"i" + hat"j" + hat"k"

vec"OQ" - 2hat"i" + 5hat"j"

vec"OR" = 3hat"i" + 2hat"j" - 3hat"k"

vec"OS" = hat"i" - 6hat"j" - hat"k"

vec"PQ" = vec"OQ" - vec"OP"

= (2hat"i" + 5hat"j") - (hat"i" + hat"j" + hat"k")

= 2hat"i" + 5hat"j" - hat"i" - hat"j" - hat"k"

vec"PQ" = hat"i" + 4hat"j" - hat"k"

vec"RS" = vec"OS" - vec"OR"

= (hat"i" - 6hat"j" - hat"k") - (3hat"i" + 2hat"j" - 3hatk")

= hat"i" - 6hat"j" - hat"k" - 3hat"i" - 2hat"j" + 3hat"k"

= -2hat"i" - 8hat"j" + 2hat"k"

= -2(hat"i" + 4hat"j" - hat"k")

vec"RS" = -2  vec"PQ"

∴ vec"RS" and vec"PQ" are parallel vectors.

Two vectors vec"a" and vec"b" are parallel vectors if vec"a" = lambdavec"b" where lambda is a scalar.

Concept: Representation of a Vector and Types of Vectors
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