Let p-q-r- be three mutually perpendicular vectors of the same magnitude If a vector x- satisfies the equation p- x-q- -p- -q- x-r- -q- -r- x-p- -r- -0 is given by

The correct option is C 1/2 ( p⃗+q⃗+r⃗ ) p ×( ( x q #160;) × p #160;) + q ×( ( x r #160;) × q #160;) + r ×( ( x p #160;) × ...

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Byju's Answer

Standard XII

Mathematics

Orthogonal System of Vectors

Let p⃗,q⃗,r...

Question

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

\frac{1}{2} (\to p + \to q - 2 \to r)

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\frac{1}{3} (\to p + \to q + \to r)

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\frac{1}{2} (\to p + \to q + \to r)

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\frac{1}{3} (2 \to p + \to q - \to r)

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Solution

The correct option is C $\frac{1}{2} (\to p + \to q + \to r)$
$\to p \times ((\to x - \to q) \times \to p) + \to q \times ((\to x - \to r) \times \to q) + \to r \times ((\to x - \to p) \times \to r) = 0 \Rightarrow (\to p . \to p) (\to x - \to q) - (\to p . (\to x - \to q) \to p) + (\to q . \to q) (\to x - \to r) - (\to q . (\to x - \to r) \to q) + \to r . \to r (\to x - \to p) - (\to r (\to x - \to p) \to r) = 0 \Rightarrow λ^{2} Σ (\to x - \to q) - Σ (\to p . \to x \to p) = 0$
Where
$\to p . \to p = \to q \to q = \to r \to r = λ^{2} (S i n c e \to p . \to q = \to q \to r = \to r \to p = 0)$
$\Rightarrow λ^{2} [3 \to x - (\to p + \to q + \to r)] - Σ {(\to p \to x) \to p} = 0 [∵ Σ (\to p \to x) \to p = \to x λ^{2}] \Rightarrow 2 \to x = \to p + \to q + \to r \Rightarrow \to x = \frac{\to p + \to q + \to r}{2}$

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Similar questions

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

Q. Let

\to p, \to q, \to r

be three mutually perpendicular vectors of the same magnitude. lf 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}

is given by

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. Three vectors

\to P

\to Q

\to R

are such that the

| \to P |

= | \to Q |

| \to R |

= \sqrt{2}

| \to P |

and

\to P + \to Q

+ \to R = 0

. The angle between

\to P

and

\to Q

\to Q

and

\to R

and

\to P

and

\to R

will be respectively.

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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Let →p,→q,→r be three mutually perpendicular vectors of the same magnitude If a vector →x satisfies the equation →p×((→x−→q)×→p)+→q×((→x−→r)×→q)+→r×((→x−→p)×→r)=0 is given by

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