Let a plane P contain two lines vec r = hat i +λ( hat i + hat j ), λ ∈ R and vec r =- hat j +μ( hat j - hat k ), μ ∈ R If Q (α, β, γ) is the foot of the perpendicular drawn from the point M (1,0,1) to P , then 3(α+β+γ) equals.

Question

Let a plane P contain two lines vec r = hat i +λ( hat i + hat j ), λ ∈ R and vec r =- hat j +μ( hat j - hat k ), μ ∈ R If Q (α, β, &gamma;) is the foot of the perpendicular drawn from the point M (1,0,1) to P , then 3(α+β+&gamma;) equals. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Dr's normal to plane =|i&j&k 1&1&0 0&1&-1|= hati+ hatj+ hatk Equation of plane -1(x-1)+1(y-0)+1(z-0)=0 x-y-z-1=0 .....(1) Now (α-1/1)=(β-0/-1)=(&gamma;-1/-1)=-((1-0-1-1)/3) (α-1/1)=(β/-1)=(&gamma;-1/-1)=(1/3) α=(4/3), β=-(1/3), &gamma;=(2/3) 3(α+β+&gamma;)=3((4/3)-(1/3)+(2/3))=5

Q. Let a plane P contain two lines r=i^+λ(i^+j^​),λ∈R and r=−j^​+μ(j^​−k^),μ∈R If Q(α,β,γ) is the foot of the perpendicular drawn from the point M(1,0,1) to P, then 3(α+β+γ) equals____.

Q. Let a plane $P$ contain two lines $r = \hat{i} + λ (\hat{i} + \hat{j}), λ \in R$ and $r = - \hat{j} + μ (\hat{j} - \hat{k}), μ \in R$
If $Q (α, β, γ)$ is the foot of the perpendicular drawn from the point $M (1, 0, 1)$ to $P$ , then $3 (α + β + γ)$ equals____.