Collinearity Of two vectors
Last updated at April 16, 2024 by Teachoo
Example 27 If π Μ + π Μ + π Μ, 2π Μ + 5π Μ, 3π Μ + 2π Μ β 3π Μ and π Μ β 6π Μ β π Μ are the position vectors of points A, B, C and D respectively, then find the angle between (π΄π΅) β and (πΆπ·) β . Deduce that (π΄π΅) β and (πΆπ·) β are collinear.Angle between (π΄π΅) β & (πΆπ·) β is given by cos ΞΈ = ((π¨π©) β . (πͺπ«) β)/|(π¨π©) β ||(πͺπ«) β | A(π Μ + π Μ + π Μ), B(2π Μ + 5π Μ) (π¨π©) β = (2 β 1) π Μ + (5 β 1) π Μ + (0 β 1) π Μ = 1π Μ + 4π Μ β π Μ |(π¨π©) β | = β(1^2+4^2+(β1)^2 ) = β18 = β(9 Γ 2) = 3βπ C(3π Μ + 2π Μ β 3π Μ), D(π Μ β 6π Μ β π Μ) (πΆπ·) β = (1 β 3) π Μ + (β6 β 2) π Μ + (β1 β (-3)) π Μ = β2π Μ β 8π Μ + 2π Μ |(πͺπ«) β | = β((β2)^2+(β8)^2+2^2 ) = β72 = β(36 Γ 2) = 6βπ Now, cos ΞΈ = ((π¨π©) β . (πͺπ«) β)/|(π¨π©) β ||(πͺπ«) β | = ((π Μ + 4π Μ β π Μ ).(β2π Μ β 8π Μ + 2π Μ ))/(3β2 Γ 6β2) = (1(β2) + 4(β8) β 1(2))/(3β2 Γ 6β2) = (β2 β 32 β 2)/(3 Γ 6 Γ β2 Γ β2) = (β36)/36 = β1 Since cos ΞΈ = β1, ΞΈ = 180Β° So, ΞΈ = 180Β° = 180Β° Γ π/180 = Ο So, angle between (π΄π΅) β & (πΆπ·) β is Ο Also, Since angle between (π΄π΅) β & (πΆπ·) β is 180Β° , they are in opposite directions Since (π΄π΅) β & (πΆπ·) β are parallel to the same line π β, they are collinear.