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15. Find the ...

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15. Find the acute angle between the lines l1 and l2 where l1 is formed by joining the points (0,0) and (2,3) and l2 by joining points (2,-2) and (3,5)

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Q. The vector equations of two lines

L_{1}

L_{2}

are respectivly

\to r = 17^i - 9^j + 9^k + λ (3^i +^j + 5^k)

\to r = 15^i - 8^j -^k + μ (4^i + 3^j)

L_{1}

L_{2}

are skew lines
II

(11, - 11, - 1)

is the point of intersection of

L_{1}

L_{2}

(- 11, - 11, 1)

is the point of intersection of

L_{1}

L_{2}

c o s^{- 1} (3 / \sqrt{35})

is the acute angle between

L_{1}

L_{2}

then, which of the following is true ?

Q. Consider the following statements relating to 3 lines

L_{1}

L_{2}

L_{3}

in the same plane
(1). If

L_{2}

L_{3}

are both parallel to

L_{1}

, then they are parallel to each other.
(2). If

L_{2}

L_{3}

are both perpendicular to

L_{1}

, then they are parallel to each other.
(3). If the acute angle between

L_{1}

L_{2}

is equal to to acute angle between

L_{1}

L_{3}

L_{2}

L_{3}

.

Of these statements:

L_{1}

L_{2}

be the lines whose equation are

\frac{x - 3}{3} = \frac{y - 8}{- 1} = \frac{z - 3}{1}

\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}

respectively A and B are two points on

L_{1}

L_{2}

respectively such that AB is perpendicular both the lines

L_{1}

L_{2}

.Find points A, B and hence find shortest distance between lines

L_{1}

L_{2}

L_{1}

L_{2}

be two intersecting lines. lf the image of

L_{1}

L_{2}

L_{2}

L_{1}

concides, then the angle between

L_{1}

L_{2}

L_{1}

L_{2}

are two lines whose vector equations are

L_{1} : \to r = λ (cos θ + \sqrt{3})^i + (\sqrt{2} sin θ)^j + (cos θ - \sqrt{3})^k

L_{2} : \to r = μ (a^i + b^j + c^k),

λ

μ

are scalars and

α

is the acute angle between

L_{1}

L_{2} .

α

is independent of

θ,

then the value of

α

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