Consider points A,B,C annd D with position vectors `7hati-4hatj+7hatk,hati-6hatj+10hatk,-1hati-3hatj+4hatk and 5hati-hatj+5hatk`, respectively. Then, ABCD is

Options

A. square
B. rhombus
C. rectangle
D. none of these

Explanation

To determine the nature of the quadrilateral formed by the points A, B, C, and D with the given position vectors, we will calculate the lengths of the sides AB, BC, CD, and DA.

Given position vectors:
- A = \(7\hat{i} - 4\hat{j} + 7\hat{k}\)
- B = \(\hat{i} - 6\hat{j} + 10\hat{k}\)
- C = \(-\hat{i} - 3\hat{j} + 4\hat{k}\)
- D = \(5\hat{i} - \hat{j} + 5\hat{k}\)

### Step 1: Calculate the length of AB

The position vector of A is \( \vec{A} = 7\hat{i} - 4\hat{j} + 7\hat{k} \) and the position vector of B is \( \vec{B} = \hat{i} - 6\hat{j} + 10\hat{k} \).

Using the distance formula:
\[ AB = |\vec{B} - \vec{A}| = |(\hat{i} - 6\hat{j} + 10\hat{k}) - (7\hat{i} - 4\hat{j} + 7\hat{k})| \]
\[ = |(-6\hat{i} + (-6 + 4)\hat{j} + (10 - 7)\hat{k})| \]
\[ = |(-6\hat{i} - 2\hat{j} + 3\hat{k})| \]
\[ = \sqrt{(-6)^2 + (-2)^2 + (3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

### Step 2: Calculate the length of BC

Now, we calculate the length of BC:
\[ BC = |\vec{C} - \vec{B}| = |(-\hat{i} - 3\hat{j} + 4\hat{k}) - (\hat{i} - 6\hat{j} + 10\hat{k})| \]
\[ = |(-\hat{i} - 3\hat{j} + 4\hat{k}) - (\hat{i} - 6\hat{j} + 10\hat{k})| \]
\[ = |(-1 - 1)\hat{i} + (-3 + 6)\hat{j} + (4 - 10)\hat{k}| \]
\[ = |(-2\hat{i} + 3\hat{j} - 6\hat{k})| \]
\[ = \sqrt{(-2)^2 + (3)^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

### Step 3: Calculate the length of CD

Next, we calculate the length of CD:
\[ CD = |\vec{D} - \vec{C}| = |(5\hat{i} - \hat{j} + 5\hat{k}) - (-\hat{i} - 3\hat{j} + 4\hat{k})| \]
\[ = |(5 + 1)\hat{i} + (-1 + 3)\hat{j} + (5 - 4)\hat{k}| \]
\[ = |(6\hat{i} + 2\hat{j} + 1\hat{k})| \]
\[ = \sqrt{(6)^2 + (2)^2 + (1)^2} = \sqrt{36 + 4 + 1} = \sqrt{41} \]

### Step 4: Calculate the length of DA

Finally, we calculate the length of DA:
\[ DA = |\vec{A} - \vec{D}| = |(7\hat{i} - 4\hat{j} + 7\hat{k}) - (5\hat{i} - \hat{j} + 5\hat{k})| \]
\[ = |(7 - 5)\hat{i} + (-4 + 1)\hat{j} + (7 - 5)\hat{k}| \]
\[ = |(2\hat{i} - 3\hat{j} + 2\hat{k})| \]
\[ = \sqrt{(2)^2 + (-3)^2 + (2)^2} = \sqrt{4 + 9 + 4} = \sqrt{17} \]

### Conclusion

We have the lengths of the sides:
- \(AB = 7\)
- \(BC = 7\)
- \(CD = \sqrt{41}\)
- \(DA = \sqrt{17}\)

Since \(AB = BC\) but \(CD \neq DA\), the quadrilateral ABCD is neither a rectangle nor a square, and the opposite sides are not equal. Thus, ABCD does not fit the criteria for any of the standard quadrilaterals.

### Final Answer
The quadrilateral ABCD is neither a rectangle nor a square, and none of the given options apply.

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