Answer

Verified

446.1k+ views

**Hint:**Check every option one by one. For example if relation \[p\overrightarrow a + q\overrightarrow b + r\overrightarrow c + s\overrightarrow d = 0\] is given then for coplanar $p + q + r + s = 0$. For the second and third use ratio formula.

**Complete step-by-step answer:**

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.

**Note:**If three points are given then they are always coplanar, to determine a plane we must need 3 points. If four points are given and they are linearly independent and the sum of their coefficients is zero then these four points lie in the same plane that means coplanar.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE