If a- and b- are adjacent side of parallelogram - then -a- - b- - -a- - b- is necessary and sufficient condition for a parallelogram to be a

The correct option is B rhombus | a⃗ +b⃗ #160; | =| a⃗ b⃗ #160; | ⇒| a⃗ #160; | #160; ^ 2 +| b⃗ #160; | #160; ^ 2 +2| a⃗ #160; | | b⃗ ...

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Byju's Answer

Standard XII

Physics

Vector Addition

If a⃗ and ...

Question

If $\to a$ and $\to b$ are adjacent side of parallelogram , then $| \to a + \to b | = | \to a - \to b |$ is necessary and sufficient condition for a parallelogram to be a

rhombus

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trapezium

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square

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rectangle

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Solution

The correct option is B rhombus
$∣ ∣ \to a + \to b ∣ ∣ = ∣ ∣ \to a - \to b ∣ ∣$
$\Rightarrow {| \to a |}^{2} + {∣ ∣ \to b ∣ ∣}^{2} + 2 | \to a | ∣ ∣ \to b ∣ ∣ cos θ = {| \to a |}^{2} + {∣ ∣ \to b ∣ ∣}^{2} - 2 | \to a | ∣ ∣ \to b ∣ ∣ cos θ$
$\Rightarrow 4 | \to a | ∣ ∣ \to b ∣ ∣ cos θ = 0$
$∵ | \to a | \neq 0 \neq ∣ ∣ \to b ∣ ∣$
$∴$ $cos θ = 0$
$\Rightarrow θ = π / 2$
Hence the parallelogram with sides $\to a & \to b$ obeying
$∣ ∣ \to a + \to b ∣ ∣ = ∣ ∣ \to a - \to b ∣ ∣$
is a rhombus
Option A.

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Similar questions

Q. Let

(\to a, \to b) = \frac{π}{3}

, if

2 \to a + \to b, \to a - 2 \to b

are the adjacent sides of a parallelogram and if

| \to a | = | \to b | = 1

, then the lengths of the diagonals are

Q. There are two vectors

\to A = 3^i +^j

and

\to B =^j + 2^k

. For these two vectors
(i) find the component of

\to A

along

\to B

and perpendicular to

\to B

in vector form.
(ii) If

\to A

and

\to B

are the adjacent sides of a parallelogram then find the magnitude of its area.
(iii) find a unit vector which is perpendicular to both

\to A

and

\to B

Q. Assertion :If

| \to a + \to b | = | \to a - \to b |

, then

\to a

and

\to b

are perpendicular to each other Reason: If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.

Q. If vectors

\to a

and

\to b

are two adjacent sides of a parallelogram then the vector representing the altitude of the parallelogram which is perpendicular to

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Q. ABCD is parallelogram. If

\to A B = \to a, \to B C = \to b

then show that

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If →a and →b are adjacent side of parallelogram , then |→a+→b|=|→a−→b| is necessary and sufficient condition for a parallelogram to be a

If $\to a$ and $\to b$ are adjacent side of parallelogram , then $| \to a + \to b | = | \to a - \to b |$ is necessary and sufficient condition for a parallelogram to be a