Home
Class 12
MATHS
Find the values of lambda such that x ,y...

Find the values of `lambda` such that `x ,y ,z!=(0,0,0)` and `( hat i+ hat j+3 hat k)x+(3 hat i-3 hat j+ hat k)y+(-4 hat i+5 hat j)z=lambda(x hat i+y hat j+z hat k),` where `hat i ,` `hat j` and `hat k` are unit vector along coordinate axes.

Text Solution

Verified by Experts

The correct Answer is:
`0,-1`
Since `(hat(i) +hat(j) + 3hat(k)) x+ (3hat(i) -3hat(j) + hat(k)) y+ (-4hat(i) +5hat(j)) z`
`=lambda (hat(i) x+ hat(j) y + hat(k) z)`
`rArr x + 3y -4z = lambda x,-3y + 5z =lambday, 3x + y+ 0 z= lambda z`
`rArr (1-lambda ) x+ 3y -4z =0, x -(3 +lambda ) y+ 5z =0`
` 3x+ y - lambda z=0`
Since `(x,y,z) ne (0,0,0)`
`:. ` Non-trivial solution
`rArr Delta =0`
`rArr |{:(1-lambda,,3,,-4),(1,,-(3+lambda),,5),(3,,1,,-lambda):}|=0`
`rArr (1-lambda)(3lambda +lambda^(2) -5) -3 (-lambda -15)-4 (1+9+3lambda) =0`
`rArr -lambda^(3) -2lambda^(3) - lambda =0 rArr lambda(lambda +1)^(2) =0`
`:. lambda=0,-1`
Promotional Banner

Topper's Solved these Questions

  • VECTOR ALGEBRA

    IIT JEE PREVIOUS YEAR|Exercise Solving Equations and Reciprocal of Vectors (Objective Questions II) (Fill in the Blanks)|4 Videos

  • TRIGONOMETRICAL RATIOS AND IDENTITIES

    IIT JEE PREVIOUS YEAR|Exercise HEIGHT AND DISTANCE|10 Videos

Similar Questions

Explore conceptually related problems

Find the values of lambda such that x,y,z!=(0,0,0)and(hat i+hat j+3hat k)x+(3hat i-3hat j+hat k)y+(-4hat i+5hat j)z=lambda(xhat i+yhat j+zhat k) where hat i,hat j,hat k are unit vector along coordinate axes.

If quad (x,y,z)!=(0,0,0) and (hat i+hat j+3hat k)x+(3hat i-3hat j+hat k)y+(-4hat i+5hat j)z=a(xhat i+yhat j+zhat k) then the values of a are

Write the value of [ hat i- hat j\ hat j- hat k\ hat k- hat i]

Find the angle between the plane hat r .( hat i - 2 hat j + 3 hat k )=5 and the line hat r = ( hat i + hat j -hat k ) + lambda ( hat i- hat j + hat k )

Evaluate hat i.(hat jxhat k)+hat j*(hat kxhat i)+hat k*(hat ixhat j)

The angle between the vectors (hat i + hat j + hat k) and ( hat i - hat j -hat k) is

The line through hat i+3 hat j+2 hat k and _|_ to the line vec r=( hat i+2 hat j- hat k)+lambda(2 hat i+ hat j+ hat k)a n d vec r=(2 hat i+6 hat j+ hat k)+mu( hat i+2 hat j+3 hat k) is a. vec r=( hat i+2 hat j- hat k)+lambda(- hat i+5 hat j-3 hat k) b. vec r= hat i+3 hat j+2 hat k+lambda( hat i-5 hat j+3 hat k) c. vec r= hat i+3 hat j+2 hat k+lambda( hat i+5 hat j+3 hat k) d. vec r= hat i+3 hat j+2 hat k+lambda(- hat i-5 hat j-3 hat k)

If vec r=(hat i+2hat j+3hat k)+lambda(hat i-hat j+hat k) and vec r=(hat i+2hat j+3hat k)+mu(hat i+hat j-hat k) are bisector of two lines is

Find the value of mu if the points with position vectors hat i - hat j + 3 hat k and 3 hat i + 4 hat j+ mu hat k are equidistant form the plane hat r .(5 hat i + 2 hat j - 7 hat k )+ 8 =0

If the vectors 3 hat i +3 hat j +9 hat k and hat i +a hat j +3 hat k are parallel, then a =