Four points with position vectors `7i-4j+7k, i -6j+10k,-i-3j+4k and 5i-j+k` form a

Options

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

Answer

To determine the shape formed by the four points with given position vectors, we need to calculate the lengths of the sides and diagonals formed by these points. The position vectors are:

1. A: \( \mathbf{a} = 7\mathbf{i} - 4\mathbf{j} + 7\mathbf{k} \)
2. B: \( \mathbf{b} = \mathbf{i} - 6\mathbf{j} + 10\mathbf{k} \)
3. C: \( \mathbf{c} = -\mathbf{i} - 3\mathbf{j} + 4\mathbf{k} \)
4. D: \( \mathbf{d} = 5\mathbf{i} - \mathbf{j} + \mathbf{k} \)

### Step 1: Find the coordinates of each point
- A: \( (7, -4, 7) \)
- B: \( (1, -6, 10) \)
- C: \( (-1, -3, 4) \)
- D: \( (5, -1, 1) \)

### Step 2: Calculate the lengths of the sides
Using the distance formula for points in 3D space:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

#### Length 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 AC:
\[ AC = \sqrt{(-1 - 7)^2 + (-3 + 4)^2 + (4 - 7)^2} = \sqrt{(-8)^2 + (1)^2 + (-3)^2} = \sqrt{64 + 1 + 9} = \sqrt{74} \]

#### Length AD:
\[ AD = \sqrt{(5 - 7)^2 + (-1 + 4)^2 + (1 - 7)^2} = \sqrt{(-2)^2 + (3)^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

#### Length 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 CD:
\[ CD = \sqrt{(5 + 1)^2 + (-1 + 3)^2 + (1 - 4)^2} = \sqrt{(6)^2 + (2)^2 + (-3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

#### Length BD:
\[ BD = \sqrt{(5 - 1)^2 + (-1 + 6)^2 + (1 - 10)^2} = \sqrt{(4)^2 + (5)^2 + (-9)^2} = \sqrt{16 + 25 + 81} = \sqrt{122} \]

#### Length AC (already calculated):
\[ AC = \sqrt{74} \]

### Step 3: Analyze the lengths
- The lengths of sides AB, AD, BC, and CD are all equal to 7.
- The diagonals AC and BD are not equal (\( \sqrt{74} \neq \sqrt{122} \)).

### Conclusion
Since all four sides are equal and the diagonals are not equal, the shape formed by these four points is a **rhombus**.