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Byju's Answer

Standard XII

Mathematics

Direction Cosines

Let L1 and L2...

Question

Let $L_{1}$ and $L_{2}$ denotes the lines $\vec{r} = \hat{i} + λ (- \hat{i} + 2 \hat{j} + 2 \hat{k}), λ \in ℝ$ and $\vec{r} = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \in ℝ$ respectively. If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them, then which of the following options describe(s) $L_{3}$ ?

$\vec{r} = \frac{1}{3} (2 \hat{i} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

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$\vec{r} = \frac{2}{9} (2 \hat{i} - \hat{j} + 2 \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

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$\vec{r} = t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

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$\vec{r} = \frac{2}{9} (4 \hat{i} + \hat{j} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

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Solution

The correct option is D
$\vec{r} = \frac{2}{9} (4 \hat{i} + \hat{j} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$

Explanation for the correct option(s):
Given that $L_{1}$ and $L_{2}$ denotes the lines $\vec{r} = \hat{i} + λ (- \hat{i} + 2 \hat{j} + 2 \hat{k}), λ \in ℝ$ and $\vec{r} = μ (2 \hat{i} - \hat{j} + 2 \hat{k}), μ \in ℝ$ respectively.
If $L_{3}$ is a line which is perpendicular to both $L_{1}$ and $L_{2}$ and cuts both of them and we need to find possible equation of $L_{3}$
Let $L_{3}$ cuts both lines $L_{1}$ and $L_{2}$ at A and B and their general points are given as
$A = (- λ + 1, 2 λ, 2 λ) B = (2 μ, - μ, 2 μ)$
so, direction ratio of line meeting A and B is $(2 μ + λ - 1, - μ - 2 λ, 2 μ - 2 λ)$
Now, direction ratio of any line which is perpendicular to lines $L_{1}$ and $L_{2}$ can be calculated as vector product of both lines $= |\begin{matrix} i & j & k \\ - 1 & 2 & 2 \\ 2 & - 1 & 2 \end{matrix}| = 6 i + 6 j - 3 k$
So direction ratios of $L_{3}$ is $(6, 6, - 3)$ comparing this with $(2 μ + λ - 1, - μ - 2 λ, 2 μ - 2 λ)$ we get,
$λ = \frac{1}{9}, μ = \frac{2}{9}$ so the required points A and B became
$(\frac{8}{9}, \frac{2}{9}, \frac{2}{9}), (\frac{4}{9}, \frac{- 2}{9}, \frac{4}{9})$
From given options, we see that for point A and B Lines $\vec{r} = \frac{2}{9} (2 \hat{i} - \hat{j} + 2 \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$ and $\vec{r} = \frac{2}{9} (4 \hat{i} + \hat{j} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$ satisfy for $L_{3}$ and If midpoint of points A and B $(\frac{8}{9}, \frac{2}{9}, \frac{2}{9}), (\frac{4}{9}, \frac{- 2}{9}, \frac{4}{9})$ which is $(\frac{2}{3}, 0, \frac{1}{3})$ satisfying for equation in option (A) as $\vec{r} = \frac{1}{3} (2 \hat{i} + \hat{k}) + t (2 \hat{i} + 2 \hat{j} - \hat{k}), t \in ℝ$
Hence, the correct options are (A), (B) and (D)

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Similar questions

Q. Let

L_{1}

, and

L_{2}

denotes the lines

\to r =^i + λ (-^i + 2^j + 2^k), λ \in R

and

\to r = μ (2^i -^j + 2^k), μ \in R

respectively.
If

L_{3}

is a line which is perpendicular to both

L_{1}

and

L_{2}

and cuts both of them, then which of the following option describe(s)

L_{3}

Q. The vector equations of the two lines

L_{1}

and

L_{2}

are given by

L_{1} : \to r = 2^i + 9^j + 13^k + λ (^i + 2^j + 3^k); L_{2} : \to r = - 3^i + 7^j + p^k + μ (-^i + 2^j - 3^k)

then the lines

L_{1}

and

L_{2}

are

Q. Consider the following statements relating to

3

lines

L_{1}, L_{2}

and

L_{3}

in the same plane

1.

L_{2}

and

L_{3}

are both parallel to

L_{1}

, then they are parallel to each other

2.

L_{2}

and

L_{3}

are both perpendicular to

L_{1}

, then they are parallel to each other

3.

If the acute angle between

L_{1}

and

L_{2}

is equal to the acute angle between

L_{1}

and

L_{3}

then

L_{2}

is parallel to

L_{3}

.
Of these statements which one is correct?

Q. Let

L_{1}

, and

L_{2}

denotes the lines

\to r =^i + λ (-^i + 2^j + 2^k), λ \in R

and

\to r = μ (2^i -^j + 2^k), μ \in R

respectively.
If

L_{3}

is a line which is perpendicular to both

L_{1}

and

L_{2}

and cuts both of them, then which of the following option describe(s)

L_{3}

Q. Line

l_{1}

is perpendicular to line

l_{2}

and line

l_{2}

is perpendicular to line

l_{3 .}

Then, what can be said about lines

l_{1}

and

l_{3 .}

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Let L1 and L2 denotes the lines r→=i^+λ-i^+2j^+2k^,λ∈ℝand r→=μ2i^-j^+2k^,μ∈ℝ respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of the following options describe(s) L3?