Test Index

Vector Algebra Part 3

Section: Mathematics

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Question : 30 of 72

Marks: +1, -0

Let

\vec{a}=\hat{i}+2\hat{j}+\hat{k}

\vec{b}=2\hat{i}+7\hat{j}+3\hat{k}

L_1: \vec{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \vec{a}, \lambda \in R

L_2: \vec{r}=(\hat{j}+\hat{k})+\mu \vec{b}, \mu \in R

be two lines. If the line

L_3

passes through the point of intersection of

L_1

L_2

and is parallel to

\vec{a}+\vec{b}

L_3

passes through the point

[29 Jan 2025 Shift 1]

$(8,26,12)$
$(2,8,5)$
$(5,17,4)$
$(-1,-1,1)$

Solution: 👈: Video Solution

\;\vec{a}=\hat{i}+2\hat{j}+\hat{k}

\;\vec{b}=2\hat{i}+7\hat{j}+3\hat{k}

\;L_1: \vec{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \vec{a}, \lambda \in R

\;\;\text{ and }\; L_2=(\hat{j}+\hat{k})+\mu \vec{b}, \mu \in R

\;\vec{a}+\vec{b}=3\hat{i}+9\hat{j}+4\hat{k}

\;L_1: \vec{r}=(\lambda-1)\hat{i}+2(\lambda+1)\hat{j}+(\lambda+1)\hat{k}

\;L_2: \vec{r}=2\mu\hat{i}+(7\mu+1)\hat{j}+(1+3\mu)\hat{k}

For point of intersection of

L_1 \& L_2

\;\lambda-1=2\mu\;\text{ and }\;2(\lambda+1)=7\mu+1

\;\Rightarrow\lambda=3\;\text{ and }\;\mu=1

\;L_3: \vec{r}=2\hat{i}+8\hat{i}+4\hat{k}+\alpha(3\hat{i}+9\hat{j}+4\hat{k})

For

\alpha=2

\vec{r}=8\hat{i}+26\hat{j}+12\hat{k}

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