Question

Using the vector method , find the values of `lambda " and " mu` for which the points
`A(3, lambda ,mu) B(2,0,-2) " and " C(1,-2,-5)` are collinear

Answer 1

Let `vec( a) , vec(b) " and " vec( c) ` be the position vectors of the given points A,B and C respectively . Then
`vec( a) =3hat(i) +lambdahat(j) + mu hat(k) , vec( b) =2hat(i) - 3hat(k) " and " vec (c ) =hat(i) -2hat(j) -5hat(k)`
Now the vector equations of te line BC is given by ,
`vec (r )= vec (b) +alpha(vec( c) -vec( b)) ` for some scalar `alpha`
`rArr vec (r ) =(1-alpha) vec( b) + alpha vec( c)`
`rArr vec(r ) =(1-alpha) (2hat(i) -3hat(k)) +alpha(hat(i) -2hat(j)-5hat(k))`
`rArr vec(r)=(2-2alpha+alpha)hat(i) -2alpha hat(j)+ (-3+3alpha-5alpha)hat(k)`
`rArrvec( r)=(2-alpha)hat(i) -2alpha hat(j) +(-3-2alpha)hat(k)`
If this line passes through the point A then we must have
`3hat(i)+lambdahat(j)+mu hat(k) =(2-alpha)hat(i)-2alpha hat(j)+(-3 -2alpha) hat(k)`
`hArr 2-alpha =3, -2alpha =lambda " and " -3-2alpha=mu`
`hArr alpha =-1 , lambda =2 " and " mu =(-3+2) =-1`
Hence `lambda=2 " and " mu=-1`