Consider points A, B, C and D with position vectors ` 7hat(i) - 4hat(j) + 7hat(k),hat(i) - 6hat(j) + 10hat(k), - hat(i) -3hat(j) + 4hat(k)` and `5hat(i) - hat(j) + 5hat(k)` respectively. Then ABCD is a -

Options

A. parallelogram but not a rhombus
B. square
C. rhombus
D. None of these

Answer

To determine the nature of the quadrilateral formed by points A, B, C, and D with the given position vectors, we will follow these steps:

### Step 1: Identify the Position Vectors
The position vectors of points A, B, C, and D are:
- A: \( \vec{A} = 7\hat{i} - 4\hat{j} + 7\hat{k} \)
- B: \( \vec{B} = \hat{i} - 6\hat{j} + 10\hat{k} \)
- C: \( \vec{C} = -\hat{i} - 3\hat{j} + 4\hat{k} \)
- D: \( \vec{D} = 5\hat{i} - \hat{j} + 5\hat{k} \)

### Step 2: Convert Position Vectors to Coordinates
We can express the position vectors in coordinate form:
- A(7, -4, 7)
- B(1, -6, 10)
- C(-1, -3, 4)
- D(5, -1, 5)

### Step 3: Calculate the Lengths of the Sides
To determine the nature of the quadrilateral, we will calculate the lengths of the sides AB, BC, CD, and DA using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

#### Length of AB:
\[ AB = \sqrt{(1 - 7)^2 + (-6 + 4)^2 + (10 - 7)^2} = \sqrt{(-6)^2 + (-2)^2 + (3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

#### Length of BC:
\[ BC = \sqrt{(-1 - 1)^2 + (-3 + 6)^2 + (4 - 10)^2} = \sqrt{(-2)^2 + (3)^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

#### Length of CD:
\[ CD = \sqrt{(5 + 1)^2 + (-1 + 3)^2 + (5 - 4)^2} = \sqrt{(6)^2 + (2)^2 + (1)^2} = \sqrt{36 + 4 + 1} = \sqrt{41} \]

#### Length of DA:
\[ DA = \sqrt{(7 - 5)^2 + (-4 + 1)^2 + (7 - 5)^2} = \sqrt{(2)^2 + (-3)^2 + (2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17} \]

### Step 4: Analyze the Lengths
From the calculations:
- \( AB = 7 \)
- \( BC = 7 \)
- \( CD = \sqrt{41} \)
- \( DA = \sqrt{17} \)

Since not all sides are equal, ABCD cannot be a rhombus or a square.

### Step 5: Check for Parallelogram
To check if ABCD is a parallelogram, we need to see if opposite sides are equal:
- \( AB \neq CD \)
- \( BC \neq DA \)

Since the opposite sides are not equal, ABCD is not a parallelogram.

### Conclusion
Since ABCD does not satisfy the conditions for being a parallelogram, rhombus, or square, we conclude that ABCD is none of these.