Let a and b be positive real numbers- Suppose PQ-a-b- and PS-a-b- are adjacent sides of a parallelogram - Let u and v be the projection vectors of w- along PQ and PS- respectively- If -u-v-w- and if the area of the parallelogram - is 8- then which of the following statements is-are TRUE-

Given, PQ=aî+bĵ PS=aî bĵ and nbsp; nbsp;w=î+ĵ ∴|u|=|w·PQ| PQ||=|(î+ĵ)·(aî+bĵ)|aî+bĵ|| ⇒|u|=a+b√(a^2+b^2) and |v|=|w·PS | PS||=|(î+ĵ)·(aî bĵ)|aî bĵ||=a b√(a ...

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

Byju's Answer

Standard IX

Mathematics

Addition of vectors

Let a and b b...

Question

Let $a$ and $b$ be positive real numbers. Suppose $- - \to P Q = a^i + b^j$ and $- \to P S = a^i - b^j$ are adjacent sides of a parallelogram $P Q R S .$ Let $\to u$ and $\to v$ be the projection vectors of $\to w =^i +^j$ along $- - \to P Q$ and $- \to P S,$ respectively. If $∣ ∣ \to u ∣ ∣ + ∣ ∣ \to v ∣ ∣ = ∣ ∣ \to w ∣ ∣$ and if the area of the parallelogram $P Q R S$ is $8,$ then which of the following statements is/are TRUE?

a + b = 4

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

a - b = 2

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

The length of the diagonal

P R

of the parallelogram

P Q R S

4

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

\to w

is an angle bisector of the vectors

- - \to P Q

and

- \to P S

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

Open in App

Solution

The correct option is C The length of the diagonal $P R$ of the parallelogram $P Q R S$ is $4$
Given, $- - \to P Q = a^i + b^j$
$- \to P S = a^i - b^j$
and $\to w =^i +^j$
$∴ ∣ ∣ \to u ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ \frac{\to w \cdot - - \to P Q}{∣ ∣ ∣ - - \to P Q ∣ ∣ ∣} ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \frac{(^i +^j) \cdot (a^i + b^j)}{∣ ∣ a^i + b^j ∣ ∣} ∣ ∣ ∣ ∣ ∣$
$\Rightarrow ∣ ∣ \to u ∣ ∣ = \frac{a + b}{\sqrt{a^{2} + b^{2}}}$

and $∣ ∣ \to v ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ \frac{\to w \cdot - \to P S}{∣ ∣ ∣ - \to P S ∣ ∣ ∣} ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ \frac{(^i +^j) \cdot (a^i - b^j)}{∣ ∣ a^i - b^j ∣ ∣} ∣ ∣ ∣ ∣ ∣ = \frac{a - b}{\sqrt{a^{2} + b^{2}}}$

Since $∣ ∣ \to u ∣ ∣ + ∣ ∣ \to v ∣ ∣ = ∣ ∣ \to w ∣ ∣,$
$\frac{(a + b) + (a - b)}{\sqrt{a^{2} + b^{2}}} = \sqrt{1^{2} + 1^{2}}$
$\Rightarrow 2 a = \sqrt{2} \sqrt{a^{2} + b^{2}}$
Squaring both sides,
$4 a^{2} = 2 (a^{2} + b^{2})$
$\Rightarrow 2 a^{2} = 2 b^{2}$
$\Rightarrow a = b \dots (1)$

Now, area of parallelogram $| - - \to P Q \times - \to P S | = 8$
$\Rightarrow 2 a b = 8 \dots (2)$
From $(1)$ and $(2)$
$a = b = 2$
and $- - \to P Q \cdot - \to P S = 0$
$∴ - - \to P Q ⊥ - \to P S$
Hence $P Q R S$ will be a rectangle.
Hence $Q (2, 2)$ and $S (2, - 2)$
$∴$ length of diagonal $Q S = 4$
So length of other diaognal i.e. $P R = 4$
and $\to w =^i +^j$ will lie on $- - \to P Q$ i.e. $2^i + 2^j$

Suggest Corrections

Similar questions

Q. If

\to p =^i +^j +^k

and

\to q = a^i + b^j + c^k

where

a, b, c \to {- 2, - 1, 0, 1, 2}

. then the number of vectors

\to q

is perpendicular to

\to a

Q. Let

\to a

and

\to b

be two non collinear unit vectors. If

\to u = \to a - (\to a \cdot \to b) \to b

and

\to v = \to a \times \to b

then

∣ ∣ \to v ∣ ∣

Q. Let

\to a =^i +^j +^k, \to b

and

\to c =^j -^k

be three vectors such that

\to a \times \to b = \to c

and

\to a \cdot \to b = 1

. If the length of projection vector of the vector

\to b

on the vector

\to a \times \to c

l

, then the value of

3 l^{2}

is equal to

Q. Let

∣ ∣ \to a ∣ ∣ = 2 \sqrt{2}, ∣ ∣ ∣ \to b ∣ ∣ ∣ = 3

and

(\to a . \to b) = \frac{π}{4}

if a parallelogram is constructed with adjacent sided

\to a - 3 \to b

and

\to a + \to b

then the longer diagonal is of length

Q. Let the position vectors of the points

P, A

and

B

\to r,^i +^j +^k

and

-^i +^k

. If

P A

is perpendicular to

P B

but

\to r

is not perpendicular to

\to r - (^j + 2^k),

then

\to r

Join BYJU'S Learning Program