Find the angle between the following pairs of lines -r -3 -2 k-2 k- and r -2 -56 k-3 -5 -4 k-

Let θ be the acute angle between the given lines, then cos θ = | b_1.b_2/|b_1||b_2| | The given lines are respectively parallel to the vectors b_1 =î ĵ 2k̂ and ...

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Byju's Answer

Standard XII

Mathematics

Angle between Two Line Segments

Find the angl...

Question

Find the angle between the following pairs of lines :
$r = 3^i +^j - 2^k + λ (^i -^j - 2^k) a n d r = 2^i -^j - 56^k + μ (3^i - 5^j - 4^k)$

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Solution

Let $θ$ be the acute angle between the given lines, then
$c o s θ = ∣ ∣ \frac{b_{1} . b_{2}}{| b_{1} | | b_{2} |} ∣ ∣$
The given lines are respectively parallel to the vectors
$b_{1} =^i -^j - 2^k a n d b_{2} = 3^i - 5^j - 4^k$
$∴ | b_{1} | = \sqrt{(1)^{2} + (- 1)^{2} + (- 2)^{2}} = \sqrt{1 + 1 + 4} = \sqrt{6} | b_{2} | = \sqrt{(3)^{2} + (- 5)^{2} + (- 4)^{2}} = \sqrt{9 + 25 + 16} = \sqrt{50} = 5 \sqrt{2} a n d b_{1} . b_{2} = (^i -^j - 2^k) . (3^i - 5^j - 4^k) = 1 \times 3 - 1 \times (- 5) - 2 \times (- 4) = 3 + 5 + 8 = 16 ∴ c o s θ = ∣ ∣ \frac{b_{1} . b_{2}}{| b_{1} | | b_{2} |} ∣ ∣ = \frac{16}{\sqrt{6} \times 5 \sqrt{2}} = \frac{16}{10 \sqrt{3}} = \frac{8}{5 \sqrt{3}} \Rightarrow θ = c o s^{- 1} (\frac{8}{5 \sqrt{3}})$

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Similar questions

Find the angle between the following pairs of lines :

$r = 2^i - 5^j +^k + λ (3^i + 2^j + 6^k) a n d r = 7^i - 6^k + μ (^i + 2^j + 2^k)$

$r = 3^i +^j - 2^k + λ (^i -^j - 2^k) a n d r = 2^i -^j - 56^k + μ (3^i - 5^j - 4^k)$

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

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

and

\to r = 7^i - 6^k + μ (^i + 2^j + 2^k)

(ii)

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

and

\to r = 2^i -^j - 56^k + μ (^3 i - 5^j - 4^k)

Q. The distance between the parallel lines

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

and

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

Q. Find the shortest distance between the lines

\to r =^i +^j + λ (2^i -^j +^k)

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

Q. Find the angle between the pair of lines given by

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

and

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

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Find the angle between the following pairs of lines : r=3^i+^j−2^k+λ(^i−^j−2^k) and r=2^i−^j−56^k+μ(3^i−5^j−4^k)

Find the angle between the following pairs of lines :
$r = 3^i +^j - 2^k + λ (^i -^j - 2^k) a n d r = 2^i -^j - 56^k + μ (3^i - 5^j - 4^k)$