Show that the lines $\bar r=(\hat i+\hat j-\hat k)+\lambda(2\hat i-2\hat j+\hat k)$ and $\bar r=(4\hat i-3\hat j+2\hat k)+\mu(\hat i-2\hat j+2\hat k)$ intersect each other.

Show that the lines bar r=(hat i+hat j-hat k)+λ(2hat i-2hat j+hat k) and bar r=(4hat i-3hat j+2hat k)+μ(hat i-

Question 144 of 145

\bar r=(\hat i+\hat j-\hat k)+\lambda(2\hat i-2\hat j+\hat k)

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2026Subjective· 3mImportance★★★★★

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Equate the parametric coordinates of the two lines and solve for $\lambda,\mu$ ; verify consistency in the third equation.

Line 1: $\bar r=(1+2\lambda)\hat i+(1-2\lambda)\hat j+(-1+\lambda)\hat k$

Line 2: $\bar r=(4+\mu)\hat i+(-3-2\mu)\hat j+(2+2\mu)\hat k$

For intersection, equate corresponding coordinates:

$1+2\lambda=4+\mu \quad\text{...(i)}$

$1-2\lambda=-3-2\mu \quad\text{...(ii)}$

$-1+\lambda=2+2\mu \quad\text{...(iii)}$

From (i): $\mu=2\lambda-3$ . Substituting into (ii):

$1-2\lambda=-3-2(2\lambda-3)=-3-4\lambda+6=3-4\lambda$

$1-2\lambda=3-4\lambda \implies 2\lambda=2 \implies \lambda=1$

$\mu=2(1)-3=-1$

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