The angle between the planes r -2 -2 k-3 and r -3 -6 -2 k-4A- Cos-116-21B- Cos-11-4C- Sin-14-21D- Cos-13-4

The correct option is A Cos^ 1(16/21 )Let a =2î ĵ=2k̂, b=3î 6ĵ+2k̂ If θ is the angle between the planes then cos θ =cos(a,b)=ab/|a||b|=6+6+4/√(4+1+4)√(9+36+4) ...

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Standard XII

Mathematics

Angle between Two Planes

The angle bet...

Question

The angle between the planes $r . (2^i -^j + 2^k) = 3$ and $r . (3^i - 6^j + 2^k) = 4$

C o s^{- 1} (\frac{16}{21})

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S i n^{- 1} (\frac{4}{21})

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C o s^{- 1} (\frac{1}{4})

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C o s^{- 1} (\frac{3}{4})

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Solution

The correct option is A $C o s^{- 1} (\frac{16}{21})$
Let $a = 2^i -^j = 2^k, b = 3^i - 6^j + 2^k$
If $θ$ is the angle between the planes then
$c o s θ = c o s (a, b) = \frac{a b}{| a | | b |} = \frac{6 + 6 + 4}{\sqrt{4 + 1 + 4} \sqrt{9 + 36 + 4}} = \frac{16}{3 \times 7} = \frac{16}{21}$

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Similar questions

Q. The angle between the planes

r . (2^i -^j + 2^k) = 3

and

r . (3^i - 6^j + 2^k) = 4

The acute angle between two lines $\to r = 2^i - 5^j +^k + α (3^i + 2^j + 6^k) a n d \to r = 7^i - 6^j +^k + β (^i + 2^j + 2^k)$ is

Q. The angle between the pair of lines given by

{\to r}_{1} =^i + 6^j + 4^k + λ (2^i -^j + 3^k)

{\to r}_{2} = 4^i + 2^j -^k + λ (^i + 2^j -^k)

(a) {sin}^{- 1} (\frac{3}{2 \sqrt{21}}) (b) {cos}^{- 1} (\frac{3}{2 \sqrt{21}})

(c) - {sin}^{- 1} (\frac{3}{2 \sqrt{21}}) (d) - {cos}^{- 1} (\frac{3}{2 \sqrt{21}})

Q. Two sides of a triangle are formed by the vectors

\to a = 3^i + 6^j - 2^k

and

\to b = 4^i -^j + 3^k

. The angles of the triangle are
1)

{cos}^{- 1} \frac{7}{\sqrt{75}}

{cos}^{- 1} (\sqrt{\frac{26}{75}})

{cos}^{- 1} \frac{3}{\sqrt{15}}

{cos}^{- 1} \frac{2}{3}

Q. 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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The angle between the planes r.(2^i−^j+2^k)=3 and r.(3^i−6^j+2^k)=4

The correct option is A Cos−1(1621)Let a=2^i−^j=2^k,b=3^i−6^j+2^k If θ is the angle between the planes then cosθ=cos(a,b)=ab|a||b|=6+6+4√4+1+4√9+36+4=163×7=1621

The angle between the planes $r . (2^i -^j + 2^k) = 3$ and $r . (3^i - 6^j + 2^k) = 4$

The correct option is A $C o s^{- 1} (\frac{16}{21})$
Let $a = 2^i -^j = 2^k, b = 3^i - 6^j + 2^k$
If $θ$ is the angle between the planes then
$c o s θ = c o s (a, b) = \frac{a b}{| a | | b |} = \frac{6 + 6 + 4}{\sqrt{4 + 1 + 4} \sqrt{9 + 36 + 4}} = \frac{16}{3 \times 7} = \frac{16}{21}$