Let A be vector parallel to the line of intersection of planes p1 and p2 through the origin- p1 is parallel to the vectors a-2-3k- and b-4-3k- and p2 is parallel to the vectors c-k- and d-3-3- The angle between A and 2-2k- is-

The correct options are B π4 D 3π4 Plane p_1 is parallel to a and b the normal to p_1 is along a×b nbsp;Plane p_ ...

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Byju's Answer

Standard XII

Mathematics

Angle between Two Planes

Let A be ve...

Question

Let $\to A$ be vector parallel to the line of intersection of planes $p_{1}$ and $p_{2}$ through the origin. $p_{1}$ is parallel to the vectors $\to a = 2^j + 3^k$ and $\to b = 4^j - 3^k$ and $p_{2}$ is parallel to the vectors $\to c =^j -^k$ and $\to d = 3^i + 3^j$ . The angle between $\to A$ and $2^i +^j - 2^k$ is:

\frac{π}{2}

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\frac{π}{4}

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\frac{π}{6}

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\frac{3 π}{4}

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Solution

The correct options are
B $\frac{π}{4}$
D $\frac{3 π}{4}$
Plane $p_{1}$ is parallel to $\to a$ and $\to b$ the normal to $p_{1}$ is along $\to a \times \to b$
Plane $p_{2}$ is parallel to $\to c$ and $\to d$ the normal to $p_{2}$ is along $\to c \times \to d$
$\to A$ is along the line of intersection of planes $p_{1}$ and $p_{2}$
$∴ \to A$ is along $(\to a \times \to b) \times (\to c \times \to d)$
$\to a \times \to b = ⎡ ⎢ ⎣ \begin{matrix} ^i & ^j & ^k 0 & 2 & 3 0 & 4 & - 3 \end{matrix} ⎤ ⎥ ⎦$
$=^i (- 6 - 12) -^j (0 - 0) +^k (0)$
$= - 18^i$
$\to c \times \to d = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 0 & 1 & - 1 3 & 3 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣$
$=^i (0 + 3) -^j (0 + 3) +^k (0 - 3)$
$= 3^i - 3^j - 3^k$
$= 3 (^i -^j -^k)$
$∴ \to A$ is along $^i \times (^i -^j -^k) =^j -^k$
The angle between $\to A$ and $2^i +^j - 2^k$ is $θ$
$cos θ = \frac{\to A}{∣ ∣ ∣ \to A ∣ ∣ ∣} . \frac{(2^i +^j - 2^k)}{3}$
$= \pm \frac{(^j -^k) (2^i +^j - 2^k)}{3 \sqrt{2}}$
$= \pm \frac{(1 + 2)}{3 \sqrt{2}} = \pm \frac{1}{\sqrt{2}}$
and $cos θ = \pm \frac{1}{\sqrt{2}}$
$\Rightarrow θ = \frac{π}{4}, \frac{3 π}{4}$

Suggest Corrections

Similar questions

Q. Let

\to A

be vector parallel to line of intersection of planes

P_{1}

and

P_{2}

through origin.

P_{1}

is parallel to the vector

2^j + 3^k

and

4^j - 3^k

and

P_{2}

is parallel to

^j -^k

and

3^i + 3^j

, then the angle between vector

\to A

and

2^i +^j - 2^k

Q. Let

\to A

be vector parallel to line of intersection of planes

P_{1}

and

P_{2}

through origin.

P_{1}

is parallel to the vectors

2^j + 3^k

and

4^j - 3^k

and

P_{2}

is parallel to

^j -^k

and

3 i + 3^j

, then the angle between vectors

¯ ¯¯ ¯ A

and

2^i +^j - 2^k

Q. Let

\to A

be vector parallel to line of intersection of planes

P_{1}

and

P_{2}

through origin.

P_{1}

is parallel to the vectors

2^j + 3^k

and

4^j - 3^k

and

P_{2}

is parallel to

^j -^k

and

3^i + 3^j,

then the angle between vector

\to A

and

2 \to i + \to j - 2^k

Q. Let

\to A

be vector parallel to line of intersection of planes

P_{1}

and

P_{2}

, Plane

P_{1}

is parallel to the vectors

2^j + 3^k

and

4^j - 3^k

and that

P_{2}

is parallel to

^j -^k

and

3^i + 3^j

then the angle between vector

\to A

and a given vector

2^i +^j - 2^k

Q. Assertion :

Consider planes

P_{1} : (\to r -^i) . (^i -^j -^k) = 0

and

P_{2} : (\to r - (2^i -^j -^k)) . (^i - 2^k) \times (2^i -^j - 3^k) = 0

and line

L : \to r = 5^i + λ (^i -^j -^k)

P_{1}

and

P_{2}

are parallel planes Reason:

L

is parallel to both

P_{1}

and

P_{2}

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Let →A be vector parallel to the line of intersection of planes p1 and p2 through the origin. p1 is parallel to the vectors →a=2^j+3^k and →b=4^j−3^k and p2 is parallel to the vectors →c=^j−^k and →d=3^i+3^j. The angle between →A and 2^i+^j−2^k is: