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

Options

A. parallelogram but not a rhombus
B. square
C. rhombus
D. rectangle

Explanation

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

### Step 1: Identify the Position Vectors
The position vectors for points A, B, C, and D are given as:
- 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: Write the Coordinates
We can extract the coordinates from the position vectors:
- A: \( (7, -4, 7) \)
- B: \( (1, -6, 10) \)
- C: \( (-1, -3, 4) \)
- D: \( (5, -1, 5) \)

### Step 3: Calculate Distances Between Points
We will calculate the distances between each pair of consecutive points using the distance formula in 3D:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

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

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

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

#### Distance 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 Distances
Now we have the distances:
- \( AB = 7 \)
- \( BC = 7 \)
- \( CD = \sqrt{41} \)
- \( DA = \sqrt{17} \)

### Conclusion
Since \( AB = BC \) but \( CD \neq DA \), the quadrilateral ABCD does not satisfy the conditions for being a parallelogram, rectangle, rhombus, or square. Thus, it belongs to none of these categories.