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

The correct option is B 12(p⃗+q⃗+r⃗) To make the problem easier, let nbsp; p = î nbsp;,q = ĵ nbsp;,r = k̂. LHS becomes nbsp; (x ĵ) î.( ...

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Standard XII

Mathematics

Orthogonal System of Vectors

Let p⃗,q⃗, ...

Question

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

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

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

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\frac{1}{3} (\to p + \to q - 2 \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 B $\frac{1}{2} (\to p + \to q + \to r)$
To make the problem easier, let $¯ ¯ ¯ p =^i, ¯ ¯ ¯ q =^j, ¯ ¯ ¯ r =^k .$
LHS becomes $(¯ ¯ ¯ x -^j) -^i . (¯ ¯ ¯ x -^j)^i + (¯ ¯ ¯ x -^k) -^j . (¯ ¯ ¯ x -^k)^j + (¯ ¯ ¯ x -^i) -^k . (¯ ¯ ¯ x -^i)^k$
$∴ 0 = 3 (a^i + b^j + c^k) -^i -^j -^k - a^i - b^j - c^k$
$∴^i +^j +^k = 2 a^i + 2 b^j + 2 c^k$
Comparing co-efficients, we get $a = b = c = \frac{1}{2}$
Thus, $¯ ¯ ¯ x = \frac{1}{2} (¯ ¯ ¯ p + ¯ ¯ ¯ q + ¯ ¯ ¯ r)$

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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 \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

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. lf a vector →x satisfies the equation →p×{(→x−→q)×→p}+→q×{(→x−→r)×→q}+→r×{(→x−→p)×→r}is given by

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