a m p ; L 1 : r → = ( − i ^ + 2 j ^ + k ^ ) + λ ( i ^ + 2 j ^ + k ^ ) a m p ; ⇒ r → = ( λ − 1 ) i ^ + 2 ( λ + 1 ) j ^ + ( λ + 1 ) k ^ a m p ; L 2 : r → = ( j ^ + k ^ ) + μ ( 2 i ^ + 7 j ^ + 3 k ^ ) a m p ; ⇒ r → = 2 μ i ^ + ( 1 + 7 μ ) j ^ + ( 1 + 3 μ ) k ^ \begin{aligned}& L_1: \overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \\& \Rightarrow \overrightarrow{\mathrm{r}}=(\lambda-1) \hat{\mathrm{i}}+2(\lambda+1) \hat{\mathrm{j}}+(\lambda+1) \hat{\mathrm{k}} \\& L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}) \\& \Rightarrow \overrightarrow{\mathrm{r}}=2 \mu \hat{\mathrm{i}}+(1+7 \mu) \hat{\mathrm{j}}+(1+3 \mu) \hat{\mathrm{k}}\end{aligned} am p ; L 1 : r = ( − i ^ + 2 j ^ + k ^ ) + λ ( i ^ + 2 j ^ + k ^ ) am p ; ⇒ r = ( λ − 1 ) i ^ + 2 ( λ + 1 ) j ^ + ( λ + 1 ) k ^ am p ; L 2 : r = ( j ^ + k ^ ) + μ ( 2 i ^ + 7 j ^ + 3 k ^ ) am p ; ⇒ r = 2 μ i ^ + ( 1 + 7 μ ) j ^ + ( 1 + 3 μ ) k ^
For point of intersection equating respective components
a m p ; a m p ; ⇒ λ − 1 = 2 μ ....... (1) a m p ; 2 ( λ + 1 ) = 1 + 7 μ ...... (2) a m p ; λ + 1 = 1 + 3 μ ..... (3) a m p ; We get \begin{aligned}&\begin{aligned}& \Rightarrow \lambda-1=2 \mu \quad\text{....... (1)}\\& 2(\lambda+1)=1+7 \mu \quad\text{...... (2)}\\& \lambda+1=1+3 \mu \quad\text{..... (3)}\end{aligned}\\&\text { We get }\end{aligned} am p ; am p ; ⇒ λ − 1 = 2 μ ....... (1) am p ; 2 ( λ + 1 ) = 1 + 7 μ ...... (2) am p ; λ + 1 = 1 + 3 μ ..... (3) am p ; We get
a m p ; ⇒ λ = 3 and μ = 1 a m p ; ⇒ a → + b → = 3 i ^ + 9 j ^ + 4 k ^ a m p ; L 3 : r → = 2 i ^ + 8 j ^ + 4 k ^ + α ( 3 i ^ + 9 j ^ + 4 k ^ ) a m p ; For α = 2 , r → = 8 i ^ + 26 j ^ + 12 k ^ \begin{aligned}& \Rightarrow \lambda=3 \text { and } \mu=1 \\& \Rightarrow \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \\& L_3: \overrightarrow{\mathrm{r}}=2 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}+\alpha(3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \\& \text { For } \alpha=2, \overrightarrow{\mathrm{r}}=8 \hat{\mathrm{i}}+26 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}\end{aligned} am p ; ⇒ λ = 3 and μ = 1 am p ; ⇒ a + b = 3 i ^ + 9 j ^ + 4 k ^ am p ; L 3 : r = 2 i ^ + 8 j ^ + 4 k ^ + α ( 3 i ^ + 9 j ^ + 4 k ^ ) am p ; For α = 2 , r = 8 i ^ + 26 j ^ + 12 k ^