If unit vectors - - are at right angles to each other and p-3-4-q-5-4r-p-q- and 2s-p-q- then

The correct options are A | r⃗+ks⃗ |= | r⃗ ks⃗ | for all real k B r⃗+s⃗ is perpendicular to r⃗ s⃗ C r⃗ is perpendicular to s⃗ Given ...

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Byju's Answer

Standard XII

Physics

Position Vector

If unit vecto...

Question

If unit vectors $^i$ & $^j$ are at right angles to each other and $\to p = 3^i + 4^j, \to q = 5^i, 4 \to r = \to p + \to q$ and $2 \to s = \to p - \to q$ then

| \to r + k \to s | = | \to r - k \to s |

for all real k

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\to r

is perpendicular to

\to s

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\to r + \to s

is perpendicular to

\to r - \to s

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| \to r | = | \to s | = | \to p | = | \to q |

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Solution

The correct options are
A $| \to r + k \to s | = | \to r - k \to s |$ for all real k
B $\to r + \to s$ is perpendicular to $\to r - \to s$
C $\to r$ is perpendicular to $\to s$
Given $\to p = 3^i + 4^j, \to q = 5^i, 4 \to r = \to p + \to q$ and $2 \to s = \to p - \to q$
$4 \to r = 8^i + 4^j \Rightarrow \to r = 2^i +^j$
$2 \to s = - 2^i + 4^j \Rightarrow \to s = -^i + 2^j$
(1). $| \to r + k \to s | = ∣ ∣ (2 - k)^i + (1 + 2 k)^j ∣ ∣ = 5 k^{2} + 5$
$| \to r - k \to s | = ∣ ∣ (2 + k)^i + (1 - 2 k)^j ∣ ∣ = 5 k^{2} + 5$
$∴ | \to r + k \to s | = | \to r - k \to s |$ for all real k.
(2) $\to r \cdot \to s = 0$
$∴ \to r$ is perpendicular to $\to s$
(3) $(\to r + \to s) \cdot (\to r - \to s) = (^i + 3^j) \cdot (3^i -^j) = 0$
$∴ \to r + \to s$ is perpendicular to $\to r - \to s$
(4) $| \to r | = | \to s | = \sqrt{5}$
$| \to p | = | \to q | = 5$
Hence, options A,B and C.

Suggest Corrections

Similar questions

Q. If

\to p

\to s

are not perpendicular to each other and

\to r \times \to p = \to q \times \to p

\to r . \to s = 0,

then

\to r =

Q. Suppose that

\to p, \to q

and

\to r

are three non-coplanar vectors in

R^{3}

. Let the components of vector of a vector

\to s

along

\to p, \to q

and

\to r

4

3

and

5

respectively. If the components of this vector

\to s

along

(- \to p + \to q + \to r)

(\to p - \to q + \to r)

and

(- \to p - \to q + \to r)

are

x, y

and

z

respectively , then the value of

2 x + y + z

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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If unit vectors ^i & ^j are at right angles to each other and →p=3^i+4^j,→q=5^i,4→r=→p+→q and 2→s=→p−→q then

If unit vectors $^i$ & $^j$ are at right angles to each other and $\to p = 3^i + 4^j, \to q = 5^i, 4 \to r = \to p + \to q$ and $2 \to s = \to p - \to q$ then