Find the angle between the lines r -2 i -5 j - k - 3 i -2 j -6 k - and r -7 i -6 k - i -2 j -2 k - - -CBSE 2014-

Let #952; be the angle between the given lines. The given lines are parallel to the vectors b1 #8594;=3i^+2j^+6k^ and b2 #8594;=i^+2j^+2k^, respectively ...

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Byju's Answer

Standard XII

Mathematics

Dot Product of Two Vectors

Find the angl...

Question

Find the angle between the lines $\vec{r} = (2 \hat{i} - 5 \hat{j} + \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})$ and $\vec{r} = 7 \hat{i} - 6 \hat{k} + μ (\hat{i} + 2 \hat{j} + 2 \hat{k})$ . [CBSE 2014]

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Solution

Let $θ$ be the angle between the given lines. The given lines are parallel to the vectors $\vec{b_{1}} = 3 \hat{i} + 2 \hat{j} + 6 \hat{k}$ and $\vec{b_{2}} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ , respectively.

So, the angle $θ$ between the given lines is given by

$\cos θ = \frac{\vec{b_{1}} . \vec{b_{2}}}{|\vec{b_{1}}| |\vec{b_{2}}|} = \frac{(3 \hat{i} + 2 \hat{j} + 6 \hat{k}) . (\hat{i} + 2 \hat{j} + 2 \hat{k})}{\sqrt{3^{2} + 2^{2} + 6^{2}} \sqrt{1^{2} + 2^{2} + 2^{2}}} = \frac{3 \times 1 + 2 \times 2 + 6 \times 2}{\sqrt{49} \sqrt{9}} = \frac{19}{21} \Rightarrow θ = \cos^{- 1} (\frac{19}{21})$

Thus, the angle between the given lines is $\cos^{- 1} (\frac{19}{21})$ .

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Similar questions

Q. Find the angle between the lines

\to r = 3 i + 2 j - 4 k + λ (i + 2 j + 2 k)

and

\to r = (5 j - 2 k) + μ (3 i + 2 j + 6 k)

Q. Find the shortest distance between the following pairs of lines whose vector equations are:

(i)

\vec{r} = 3 \hat{i} + 8 \hat{j} + 3 \hat{k} + λ (3 \hat{i} - \hat{j} + \hat{k}) and \vec{r} = - 3 \hat{i} - 7 \hat{j} + 6 \hat{k} + μ (- 3 \hat{i} + 2 \hat{j} + 4 \hat{k})

(ii)

\vec{r} = (3 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} - 2 \hat{j} + 7 \hat{k}) and \vec{r} = - \hat{i} - \hat{j} - \hat{k} + μ (7 \hat{i} - 6 \hat{j} + \hat{k})

(iii)

\vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) and \vec{r} = (2 \hat{i} + 4 \hat{j} + 5 \hat{k}) + μ (3 \hat{i} + 4 \hat{j} + 5 \hat{k})

(iv)

\vec{r} = (1 - t) \hat{i} + (t - 2) \hat{j} + (3 - t) \hat{k} and \vec{r} = (s + 1) \hat{i} + (2 s - 1) \hat{j} - (2 s + 1) \hat{k}

(v)

\vec{r} = (λ - 1) \hat{i} + (λ + 1) \hat{j} - (1 + λ) \hat{k} and \vec{r} = (1 - μ) \hat{i} + (2 μ - 1) \hat{j} + (μ + 2) \hat{k}

(vi)

\vec{r} = (2 \hat{i} - \hat{j} - \hat{k}) + λ (2 \hat{i} - 5 \hat{j} + 2 \hat{k}) and, \vec{r} = (\hat{i} + 2 \hat{j} + \hat{k}) + μ (\hat{i} - \hat{j} + \hat{k})

(vii)

\vec{r} = (\hat{i} + \hat{j}) + λ (2 \hat{i} - \hat{j} + \hat{k}) and, \vec{r} = 2 \hat{i} + \hat{j} - \hat{k} + μ (3 \hat{i} - 5 \hat{j} + 2 \hat{k})

(viii)

\vec{r} = (8 + 3 λ) \hat{i} - (9 + 16 λ) \hat{j} + (10 + 7 λ) \hat{k}

and

\vec{r} = 15 \hat{i} + 29 \hat{j} + 5 \hat{k} + μ (3 \hat{i} + 8 \hat{j} - 5 \hat{k})

[NCERT EXEMPLAR]

Q. Find the angle between the pair of lines

\to r = 3 i + 2 j - 4 k + λ (i + 2 j + 2 k)

and

\to r = 5 i - 2 k + μ (3 i + 2 j + 6 k)

Q. Find the angle between the line

\to r = (\to i + 2 \to j - \to k) + μ (2 \to i + \to j + 2 \to k)

and the plane

\to r . (3 \to i - 2 \to j + 6 \to k) = 0

Q. Find the angle between the following pairs of lines:
(i)

\vec{r} = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 2 \hat{k}) and \vec{r} = \hat{i} - \hat{j} + 2 \hat{k} - μ (2 \hat{i} + 4 \hat{j} - 4 \hat{k})

(ii)

\vec{r} = (3 \hat{i} + 2 \hat{j} - 4 \hat{k}) + λ (\hat{i} + 2 \hat{j} + 2 \hat{k}) and \vec{r} = (5 \hat{j} - 2 \hat{k}) + μ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})

(iii)

\vec{r} = λ (\hat{i} + \hat{j} + 2 \hat{k}) and \vec{r} = 2 \hat{j} + μ \{(\sqrt{3} - 1) \hat{i} - (\sqrt{3} + 1) \hat{j} + 4 \hat{k}\}

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Find the angle between the lines r→=2i^-5j^+k^+λ3i^+2j^+6k^ and r→=7i^-6k^+μi^+2j^+2k^. [CBSE 2014]

Find the angle between the lines $\vec{r} = (2 \hat{i} - 5 \hat{j} + \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 6 \hat{k})$ and $\vec{r} = 7 \hat{i} - 6 \hat{k} + μ (\hat{i} + 2 \hat{j} + 2 \hat{k})$ . [CBSE 2014]