A non-zero vector a- is parallel to the line of intersection of plane p 1 determined by - - and - and plane P 2 determined by vectors - - and - -k- then angle between a and vector - -2- -2k- is

The correct options are A π4 D 3π4 #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; #10; ...

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Byju's Answer

Standard XII

Mathematics

Family of Planes Passing through the Intersection of Two Planes

A non-zero ve...

Question

A non-zero vector $\to a$ is parallel to the line of intersection of plane $p_{1}$ determined by $^i +^j$ and $^i$ and plane $P_{2}$ determined by vectors $^i -^j$ and $^i +^k$ , then angle between a and vector $^i - 2^j + 2^k$ is

\frac{π}{4}

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\frac{π}{2}

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\frac{π}{3}

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\frac{3 π}{4}

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Solution

The correct options are
A $\frac{π}{4}$
D $\frac{3 π}{4}$

Equation of the plane containing $^i$ and $^i +^j$ is

$[(\to r -^i)^i (^i +^j)] = 0$

$\Rightarrow (\to r -^i) . [^i \times (^i +^j)] = 0$

$⇒{(x^i+y^j+z^k)−^i}.[^i×^i+^i×^j]=0$

$⇒{(x−1)^i+y^j+z^k}.[^k]=0⇒(x−1)^i.^k+y^j.^k+z^k.^k=0$

$\Rightarrow z = 0$ ...(i)

Equation of the plane containing $^i -^j$ and $^i +^k$ is

$[\to (r - (^i -^j)) (^i -^j) (^i +^k)] = 0$

$\Rightarrow (\to r -^i +^j) . [(^i -^j) \times (^i +^k)] = 0$

$⇒{(x^i+y^j+z^k)−(^i−^j)}.[^i×^i+^i×^k−^j×^i−^j×^k]=0$

$⇒{(x−1)^i+(y+1)^j+z^k}.[−^j+^k−^i]=0$

$\Rightarrow - (x - 1) - (y + 1) + z = 0 . . . (i i)$

Let $\to a = a_{1}^i + a_{2}^j + a_{3}^k$

Since, $\to a$ is parallel to Eqs. (i) and (ii), we obtain

$a_{3} = 0$

And $a_{1} + a_{2} - a_{3} = 0$ $\Rightarrow a_{1} = - a_{2}, a_{3} = 0$

Thus, vector in the direction $\to a$ is $^i -^j$

If $θ$ is the angle between $\to a$ and $^i - 2^j + 2^k$ , then

$cos θ = \pm \frac{(1) (1) + (- 1) (- 2)}{\sqrt{1 + 1} \sqrt{1 + 4 + 4}} = \pm \frac{3}{\sqrt{2} .3}$

$\Rightarrow cos θ = \pm \frac{1}{\sqrt{2}} \Rightarrow θ = \frac{π}{4} o r \frac{3 π}{4}$

Suggest Corrections

Similar questions

Q. A non-zero vector

\to a

is parallel to the line of intersection of the plane determined by the vectors

^i,^i +^j

and the plane determined by the vectors

^i -^j,^i +^k

. The angle between

\to a

and

^i - 2^j + 2^k

Q. A non-zero vector

\to a

is parallel to the line of intersection of the plane determined by the vectors

^i,^i,^j

and the plane determined by the vectors

^i -^j,^i -^k

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^i - 2^j + 2^k

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\to a

is parallel to the line of intersection of the plane

P_{1}

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^i +^j

and

^i - 2^j

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2^i +^j a n d 3^i + 2^k

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\to a

and vector

^i - 2^j + 2^k

Q. A non-zero vector

\to a

is parallel to the line of intersection of the plane determined by the vectors

^i,^i +^j

and the plane determined by the vectors

^i -^j,^i +^k .

If the acute angle between

\to a

and the vector

^i - 2^i + 2^k

θ

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\sqrt{2} cos θ

Q. A vector of magnitude

\sqrt{2}

coplanar with the vectors

\to a =^i +^j + 2^k

and

\to b =^i + 2^j +^k

and perpendicular to the vector

\to c =^i +^j +^k

is
(1)

-^i -^k

(2)

^j -^k

(3)

^i -^j

(4)

^i +^k

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A non-zero vector →a is parallel to the line of intersection of plane p1 determined by ^i+^j and ^i and plane P2 determined by vectors ^i−^j and ^i+^k, then angle between a and vector ^i−2^j+2^k is

A non-zero vector $\to a$ is parallel to the line of intersection of plane $p_{1}$ determined by $^i +^j$ and $^i$ and plane $P_{2}$ determined by vectors $^i -^j$ and $^i +^k$ , then angle between a and vector $^i - 2^j + 2^k$ is