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Question

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a) A,B,C,D are coplanar vectors

b) The line joining the points B and D divide the line joining points A and C in the ratio 2:1

c) The line joining the points A and C divide the line joining points B and D in the ratio 3:1

d) The four vectors $\overrightarrow a \,,\,\overrightarrow b \,,\,\overrightarrow c \,\& \,\overrightarrow d $ are linearly dependent

Answer
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Here according to the question there are four points A, B, C, D whose position vectors are $\overrightarrow a \,,\,\overrightarrow b \,,\,\overrightarrow c \,\& \,\overrightarrow d $ respectively.

i) So firstly let us check the first option where A, B, C, D are coplanar vectors.

So here we are given the relation $3\overrightarrow a - 2\overrightarrow b + \overrightarrow c - 2\overrightarrow d = 0$, and we need to check coplanarity that means whether all four points are in one plane or not.

So if we are given equation \[p\overrightarrow a + q\overrightarrow b + r\overrightarrow c + s\overrightarrow d = 0\] where a,b,c,d are position vectors then if coefficient sum will become zero, then its lies in a single plane.

Here relation is given by $3\overrightarrow a - 2\overrightarrow b + \overrightarrow c - 2\overrightarrow d = 0$

So here the sum of coefficients is $3 - 2 + 1 - 2 = 0$

Sp A, B, C, D are coplanar.

Option A is correct.

Now let's check other option

Here it is given that

$3\overrightarrow a - 2\overrightarrow b + \overrightarrow c - 2\overrightarrow d = 0$

Upon arranging we get by $3\overrightarrow a + \overrightarrow c = 2\overrightarrow b + 2\overrightarrow d $

Now if we divide on both sides by four

\[\dfrac{{3\overrightarrow a + \overrightarrow c }}{4} = \dfrac{{2\overrightarrow b + 2\overrightarrow d }}{4}\]

We can write as \[\dfrac{{3\overrightarrow a + \overrightarrow c }}{{3 + 1}} = \dfrac{{2\overrightarrow b + 2\overrightarrow d }}{{2 + 2}}\]

Here AC is divided into 1:3 ratio where BD is divided into 2:2 or 1:1 ratio.

ii) So the second option is wrong because it is saying that points B and D divide the line joining points A and C in the ratio 2:1 but here we get that BD divides AC in 1:3 ratio.

iii) Now, C option is correct as AC divides BD in 1:1 Ratio.

iv) If $\overrightarrow a \,,\,\overrightarrow b \,,\,\overrightarrow c \,\& \,\overrightarrow d $ are linearly independent then \[p\overrightarrow a + q\overrightarrow b + r\overrightarrow c + s\overrightarrow d = 0\], if it is in this form where p, q, r and s are integers then they are linearly independent. So option D is incorrect.

So option A and C are correct.