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Mathematics Question on Vectors

Let L1L_1 and L2L_2 denote the lines r⃗=i^+λ(−i^+2j^+2k^),λ ∈ R\vec{r}=\hat{i}+\lambda\left(-\hat{i}+2\hat{j}+2\hat{k}\right), \lambda\,\in\,\mathbb{R} and r⃗=μ(2i^−j^+2k^),μ ∈ R\vec{r}=\mu\left(2\hat{i}-\hat{j}+2\hat{k}\right), \mu\,\in\,\mathbb{R} respectively. If L3L_3 is a line which is perpendicular to both L1L_1 and L2L_2 and cuts both of them, then which of the following options describe(s) L3L_3?

A

r⃗=29(4i^+j^+k^)+t(2i^+2j^−k^),t ∈ R\vec{r}=\frac{2}{9}\left(4\hat{i}+\hat{j}+\hat{k}\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\,\in\,\mathbb{R}

B

r⃗=29(2i^−j^+2^k)+t(2i^+2j^−k^),t ∈ R\vec{r}=\frac{2}{9}\left(2\hat{i}-\hat{j}+\hat2{k}\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\,\in\,\mathbb{R}

C

r⃗=13(2i^+k^)+t(2i^+2j^−k^),t ∈ R\vec{r}=\frac{1}{3}\left(2\hat{i}+\hat{k}\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\,\in\,\mathbb{R}

D

r⃗=t(2i^+j^−k^),t ∈ R\vec{r}=t\left(2\hat{i}+\hat{j}-\hat{k}\right),t\,\in\,\mathbb{R}

Answer

r⃗=13(2i^+k^)+t(2i^+2j^−k^),t ∈ R\vec{r}=\frac{1}{3}\left(2\hat{i}+\hat{k}\right)+t\left(2\hat{i}+2\hat{j}-\hat{k}\right),t\,\in\,\mathbb{R}

Explanation

Solution

∵\because L3L_3 is perpendicular to both L1L_1 and L2L_2. Then a vector along L3L_3 will be, ∣i^j^k^ −122 2−12∣=3(2i^+2j^−k^)\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\\ -1&2&2\\\ 2&-1&2\end{vmatrix}=3\left(2\hat{i}+2\hat{j}-\hat{k}\right) Consider a point on L1asP(−λ+1,2λ,2μ−2λ)L_{1} as P\left(-\lambda+1,2\lambda,2\mu-2\lambda\right) and a point on L2L_2 as Q(2μ,−μ,2μ)Q\left(2\mu,-\mu, 2\mu\right) DR's of L3<2μ+λ−1,−μ−2λ,2μ−2λ>L_{3} <2\mu+\lambda-1, -\mu-2\lambda, 2\mu-2\lambda > Here 2μ+λ−12=−μ−2λ2=2μ−2λ−1\frac{2\mu+\lambda-1}{2}=\frac{-\mu-2\lambda}{2}=\frac{2\mu-2\lambda}{-1} ⇒λ=19\Rightarrow \lambda=\frac{1}{9} and μ=29\mu=\frac{2}{9} P(89,29,29)P\left(\frac{8}{9}, \frac{2}{9}, \frac{2}{9}\right) and Q(49,−29,49);Q\left(\frac{4}{9}, -\frac{2}{9}, \frac{4}{9}\right); mid-point of PQPQ is R(23,0,13)R\left(\frac{2}{3}, 0, \frac{1}{3}\right) Equation of L3:r⃗=a⃗+λ(2i^+2j^−k^),L_{3} : \vec{r}=\vec{a}+\lambda\left(2\hat{i}+2\hat{j}-\hat{k}\right), here aˉ\bar{a} is the position vector of any point on L3L_{3}. Possible vector of aˉ\bar{a} are (89i^+29j^+29k^)or(49i^−29j^+49k^)or(23i^+13k^)\left(\frac{8}{9}\hat{i}+\frac{2}{9}\hat{j}+\frac{2}{9}\hat{k}\right)or\left(\frac{4}{9}\hat{i}-\frac{2}{9}\hat{j}+\frac{4}{9}\hat{k}\right)or\left(\frac{2}{3}\hat{i}+\frac{1}{3}\hat{k}\right)