Let P 3-2-6 be a point in space and Q be a point on the line r- i-j-2k- - -3i-j-5k- Then the value of - for which the vector PQ is parallel to x-4y-3z-1 is

The correct option is B 1/4 Let O be the origin, as Q lies on the given line, OQ=i⃗ j⃗+2k⃗+μ ( 3i⃗+j⃗+5k⃗ ) #160;Also #160; OP=3 ...

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Standard XII

Mathematics

Dot Product

Let P 3,2,6...

Question

Let $P (3, 2, 6)$ be a point in space and $Q$ be a point on the line
$\to r = (\to i - \to j + 2 \to k) + μ (- 3 \to i + \to j + 5 \to k)$
Then the value of $μ$ for which the vector $P Q$ is parallel to $x - 4 y + 3 z = 1$ is

\frac{1}{4}

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- \frac{1}{4}

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\frac{1}{8}

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- \frac{1}{8}

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Solution

The correct option is B $\frac{1}{4}$
Let $O$ be the origin, as $Q$ lies on the given line,
$O Q = \to i - \to j + 2 \to k + μ (- 3 \to i + \to j + 5 \to k)$
Also $O P = 3 \to i + 2 \to j + 6 \to k$ .
As $P Q$ is parallel to the plane $x - 4 y + 3 z = 1$ .
$P Q \cdot (\to i - 4 \to j + 3 \to k) = 0$
$\Rightarrow$ $(O Q - O P) \cdot (\to i - 4 \to j + 3 \to k) = 0$
$\Rightarrow$ $(1 + 4 + 6) + μ (- 3 - 4 + 15) - (3 - 8 + 18) = 0$
$\Rightarrow$ $11 + 8 μ - 13 = 0$
$\Rightarrow μ = \frac{1}{4}$
Ans: A

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Similar questions

Q. Let

P (3, 2, 6)

be a point in space and

Q

be a point on the line

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

, then the value of

μ

for which the vector

\to P Q

is parallel to the plane

x - 4 y + 3 z = 1

Q. Let

P (3, 2, 6)

be point in space and

Q

be appoint on the line

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

. Then the value of

μ

for which the vector

\to P Q

is parallel to the plane

x - 4 y + 3 z = 1

Q. Let A be a point on the line

\to r = (1 - 3 μ)^i + (μ - 1)^j + (2 + 5 μ)^k

and

B (3, 2, 6)

be a point in the space. Then the value of

μ

for which the vector

¯ ¯¯¯¯¯¯ ¯ A B

is parallel to the plane

x - 4 y + 3 z = 1

is?

Q. Let

A

be a point on the line

\to r = (1 - 3 μ)^i + (μ - 1)^j + (2 + 5 μ)^k

and

B (3, 2, 6)

be a point in the space. Then the value of

μ

for which the vector

- - \to A B

is parallel to the plane

x - 4 y + 3 z = 1

is :

Q. If

\to r = 3 \to i + 2 \to j - 5 \to k

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

\to b = \to i + 3 \to j - 2 \to k

and

\to c = - 2 \to i + \to j - 3 \to k

such that

\to r = λ \to a + μ \to b + ν \to c

then

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Let P(3,2,6) be a point in space and Q be a point on the line→r=(→i−→j+2→k)+μ(−3→i+→j+5→k)Then the value of μ for which the vector PQ is parallel to x−4y+3z=1 is

Let $P (3, 2, 6)$ be a point in space and $Q$ be a point on the line
$\to r = (\to i - \to j + 2 \to k) + μ (- 3 \to i + \to j + 5 \to k)$
Then the value of $μ$ for which the vector $P Q$ is parallel to $x - 4 y + 3 z = 1$ is