If `veca,vecb,vecc & vecd` are position vector of point A,B,C and D respectively in 3-D space no three of A,B,C,D are colinear and satisfy the relation `3veca-2vecb+vecc-2vecd=0` then (a) A, B, C and D are coplanar (b) The line joining the points B and D divides the line joining the point A and C in the ratio of `2:1` (c) The line joining the points A and C divides the line joining the points B and D in the ratio of `1:1` (d)The four vectors `veca, vecb, vec c and vecd` are linearly dependent .

Options

A. A, B, C and D are coplanar
B. The line joining the points B and D divides the line joining the point A and C in the ratio of `2:1`
C. The line joining the points A and C divides the line joining the points B and D in the ratio of `1:1`
D. The four vectors `veca, vecb, vec c and vecd` are linearly dependent .

Answer

To solve the problem, we start with the given relation among the position vectors of points A, B, C, and D:

\[ 3\vec{a} - 2\vec{b} + \vec{c} - 2\vec{d} = 0 \]

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate the terms involving vectors A, C, B, and D:

\[ 3\vec{a} + \vec{c} = 2\vec{b} + 2\vec{d} \]

### Step 2: Dividing by 4
Next, we divide the entire equation by 4 to simplify it:

\[ \frac{3\vec{a} + \vec{c}}{4} = \frac{2\vec{b} + 2\vec{d}}{4} \]

This simplifies to:

\[ \frac{3\vec{a} + \vec{c}}{4} = \frac{1}{2}(\vec{b} + \vec{d}) \]

### Step 3: Interpreting the Result
The left-hand side represents a point that divides the line segment joining points A and C in the ratio of 3:1. The right-hand side represents a point that divides the line segment joining points B and D in the ratio of 1:1.

### Step 4: Conclusion about Ratios
From the above interpretation:
- The line joining points A and C is divided by the point in the ratio 3:1.
- The line joining points B and D is divided by the same point in the ratio 1:1.

### Step 5: Checking the Options
Now, we can evaluate the options provided:

(a) **A, B, C, and D are coplanar.**
- Since the two lines intersect at a common point, they lie in the same plane. Thus, this option is correct.

(b) **The line joining points B and D divides the line joining points A and C in the ratio of 2:1.**
- This is incorrect as we found the ratio to be 3:1.

(c) **The line joining points A and C divides the line joining points B and D in the ratio of 1:1.**
- This is correct because the point divides the line BD in the ratio of 1:1.

(d) **The four vectors are linearly dependent.**
- Since we have a non-trivial linear combination of the vectors equating to zero, this option is also correct.

### Final Answer
The correct options are (a), (c), and (d).