Let a- b- c be three distinct positive real numbers- If p- q- r- lie in a plane- where p-ai-aj-bk- q-i-k- and r-ci-cj-bk- then b is

The correct option is D HM of a, c The condition for coplanarity of vectors #160; p⃗, #160;q⃗, #160;r⃗ #160;gives #160; a #160; ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Arithmetic Progression

Let a, b, c...

Question

Let $a, b, c$ be three distinct positive real numbers. If $\to p, \to q, \to r$ lie in a plane, where $\to p = a \to i - a \to j + b \to k$ , $\to q = \to i + \to k$ and $\to r = c \to i + c \to j + b \to k$ , then $b$ is

A M

a, c

No worries! We‘ve got your back. Try BYJU‘S free classes today!

G M

a, c

No worries! We‘ve got your back. Try BYJU‘S free classes today!

H M

a, c

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Equal to

0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is D $H M$ of $a, c$
The condition for coplanarity of vectors $\to p, \to q, \to r$ gives
$∣ ∣ ∣ ∣ \begin{matrix} a & - a & b 1 & 0 & 1 c & c & b \end{matrix} ∣ ∣ ∣ ∣ = 0$

$\Rightarrow - a c + a (b - c) + b (c) = a b - 2 a c + b c = 0$

Dividing throughout by $a b c,$ we have
$\frac{1}{c} + \frac{1}{a} = \frac{2}{b}$

$\Rightarrow a, b, c$ are in $H . P .$

Suggest Corrections

Similar questions

Q. Let

a, b

and

c

be distinct non negative numbers. If the vectors

a \to i + a \to j + c \to k, \to i + \to k

and

c \to i + c \to j + b \to k

lie in a plane then

c

Q. If

[\to p + 2 \to q + 3 \to r \to q + 2 \to r + 3 \to p \to r + 2 \to p + 3 \to q] = 54

where

\to p

\to q

and

\to r

are three vector then the Value of

∣ ∣ ∣ ∣ \begin{matrix} \to p . \to p & \to p . \to q & \to p . \to r \to p . \to q & \to q . \to q & \to q . \to r \to p . \to r & \to r . \to q & \to r . \to r \end{matrix} ∣ ∣ ∣ ∣

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

)

Q. Let

\to p, \to q, \to r

be three mutually perpendicular vectors of the same magnitude If a vector

\to x

satisfies the equation

\to p \times ((\to x - \to q) \times \to p) + \to q \times ((\to x - \to r) \times \to q) + \to r \times ((\to x - \to p) \times \to r) = 0

is given by

Q. Let

\to p, \to q, \to r

be three mutually perpendicular vectors of the same magnitude. If a vector

\to x

satisfies the equation

\to p [(\to x - \to q) \times \to p] + \to q \times [(\to x - \to r) \times \to q] + \to r \times [(\to x - \to p) \times \to r] = \to 0

, then

\to x

is given by

Join BYJU'S Learning Program

Let a,b,c be three distinct positive real numbers. If →p,→q,→r lie in a plane, where →p=a→i−a→j+b→k, →q=→i+→k and →r=c→i+c→j+b→k, then b is

The correct option is D HM of a,cThe condition for coplanarity of vectors →p,→q,→r gives ∣∣ ∣∣a−ab101ccb∣∣ ∣∣=0⇒−ac+a(b−c)+b(c)=ab−2ac+bc=0Dividing throughout by abc, we have 1c+1a=2b⇒a,b,c are in H.P.

Let $a, b, c$ be three distinct positive real numbers. If $\to p, \to q, \to r$ lie in a plane, where $\to p = a \to i - a \to j + b \to k$ , $\to q = \to i + \to k$ and $\to r = c \to i + c \to j + b \to k$ , then $b$ is