Question

4.
Four vectors PQ, QR, PS and SR are connected
as shown in figure, such that S is the mid-point of
PR and |PQI = IQR), then which two vectors are
equal?

S
RA
Р
(1) PQ and QR
(2) PS and SR
(3) PQ and PS
(4) Both (1) & (2)​

Answer 1

The correct option is 2, i.e., PS and SR.

Given:

PQ, QR, PS and SR arte 4 vectors.

|PQI = IQR|

S is the mid point of PR.

To find:

The two equal vectors.

Solution:

Join QS and make it a vector.

Since, the 4 vectors are arranged in the form of a triangle. So, we can apply triangle law of vector addition.

According to triangle law of vector addition, the vector of the this side is equal to the sum of vectors of first two sides.

On applying triangle law of vector addition in ΔQSR we get,

QR= SR+QS

|QR|= $\sqrt{(SR^{2} )+(QS^{2} )}$              ...........................(1)

Similarly, on applying triangle law of vector addition in ΔQSR we get,

PQ= PS+QS

|PQ|=  $\sqrt{(PS^{2} )+(QS^{2} )}$               .............................(2)

Since, |PQI = IQR|

So,  $\sqrt{(SR^{2} )+(QS^{2} )}$=  $\sqrt{(PS^{2} )+(QS^{2} )}$

${(SR^{2} )+(QS^{2} )}$=  ${(PS^{2} )+(QS^{2} )}$

${(SR^{2})}$=  ${(PS^{2} )}$

SR= PS

Therefore, SR and PS are equal.

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Answer 2

Answer:

PS and SR

Explanation:

As per the data given in the question,

We have to determine that among the given vectors in the options which of the following vectors are equal in context to the information given in the image.

As per data given,

It is given that,

PQ, QR, PS and SR are 4 vectors.

As it is also given that,

|PQI = IQR| and

S is the mid point of PR.

For finding that which of the following vectors are equal,

We will first divide the given triangle into 2 triangles by simply joining QS as a vector.

Now, applying the triangle law of vector addition in ΔQSR

We will get the following:

PQ= PS+QS

|PQ|= $\sqrt{(PS)^{2}+(QS)^{2} }$                 ---(1)

Now applying the triangle law of vector addition in ΔQSR

We will get the following:

QR= SR+QS

|QR|= $\sqrt{(SR)^{2}+(QS)^2}$                 ---(2)

Now as it is given that |PQ|=|QR|

So, from the above-mentioned statements,

We can conclude that,

Both the above-obtained equations will be equal.

So, (1)=(2)

$\sqrt{(SR)^{2}+(QS)^2} = \sqrt{(PS)^{2}+(QS)^{2} }\\\\\\{(SR)^{2}+(QS)^2={(PS)^{2}+(QS)^{2}\\\\\\(SR)^{2}=(PS)^{2}\\SR=PS$

Hence, SR and PS are equal.

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