Pp- and Qq- are the position vectors of two fixed points and Rr- is the position vector of a variable point- If R moves such that r- - p- - r- - q- - 0- then the locus of R is

The correct option is D A line passing through points P and Q (r⃗ p⃗)×(r⃗ q⃗)=0 r⃗×(r⃗ q⃗) p⃗×(r⃗ q⃗)=0 r⃗×r⃗ r⃗×q⃗ p⃗×r⃗+p⃗×q⃗=0 ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

Pp⃗ and Qq⃗...

Question

$P (\to p)$ and $Q (\to q)$ are the position vectors of two fixed points and $R (\to r)$ is the position vector of a variable point. If $R$ moves such that $(\to r - \to p) \times (\to r - \to q) = \to 0$ , then the locus of $R$ is

A plane containing the origin

O

and parallel to two non-collinear vectors

\to O P

and

\to O Q

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The surface of a sphere described on

P Q

as its diameter

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A line passing through points

P

and

Q

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A set of lines parallel to line

P Q

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Solution

The correct option is D A line passing through points $P$ and $Q$
$(\to r - \to p) \times (\to r - \to q) = 0$
$\to r \times (\to r - \to q) - \to p \times (\to r - \to q) = 0$
$\to r \times \to r - \to r \times \to q - \to p \times \to r + \to p \times \to q = 0$
$\to p \times \to q - \to r \times \to q - \to p \times \to r = 0$
$\to p \times \to q - \to r \times \to q + \to r \times \to p = 0$
$\to p \times \to q + \to r \times (\to p - \to q) = 0$
$\to p \times \to q = \to r \times (\to q - \to p)$
Therefore
$\to r = \to p + t (\to q - \to p)$
This represents a line passing through P and Q.

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Similar questions

Q. Consider three vectors

\to P = ˆ i + ˆ j + ˆ k; \to q = 2 ˆ i + 4 ˆ j - ˆ k

and

\to r = 2 ˆ i + 4 ˆ j + 3 ˆ k

. If

\to p, \to q

and

\to r

denotes the position vector of three non-collinear points, then the equation of the plane containing these points is

Q. Consider three vectors

\to p = i + j + k, \to q = 2 i + 4 j - k

and

\to r = i + j + 3 k

. If

p, q

and

r

denotes the position vector of three non-collinear points, then the equation of the plane containing these points is

Q. A line L

_{1}

passing through a point with position vector

\to p = \to i + 2 \to j + 3 \to k

and parallel to

\to a = \to i + 2 \to j + 3 \to k

, Another line

L

_{2}

passing through a point with position vector

\to q = 2 \to i + 3 \to j + \to k

and parallel to

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

The minimum distance of origin from the plane passing through the point with position vector

\to p

and perpendicular to the line

L

_{2}

Q. The position vectors of

P

and

Q

are respectively

a

and

b

. If

R

is a point on

P Q

P Q

such that

P R = 5 P Q

, then the position vector of

R

Q. lf

\to P \times \to Q = \to R; \to Q \times \to R = \to P

and

\to R \times \to P = \to Q

are 3 non-zero vectors, then,
a)

\to P, \to Q

and

\to R

are coplanar
b) Angle between

\to P

and

\to Q

may be less than

90^{0}

\to P + \to Q + \to R

cannot be equal to zero
d)

\to P, \to Q

and

\to R

are mutually perpendicular
( Given

P \neq Q \neq R \neq 0

)

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P(→p) and Q(→q) are the position vectors of two fixed points and R(→r) is the position vector of a variable point. If R moves such that (→r−→p)×(→r−→q)=→0, then the locus of R is

$P (\to p)$ and $Q (\to q)$ are the position vectors of two fixed points and $R (\to r)$ is the position vector of a variable point. If $R$ moves such that $(\to r - \to p) \times (\to r - \to q) = \to 0$ , then the locus of $R$ is