The magnitude of a given vector with end points`(4, -4,0)` and `(-2, -2,0)` must be

Options

A. 6
B. `5sqrt(2)`
C. 4
D. `2sqrt(10)`

Explanation

To find the magnitude of a vector defined by its endpoints, we can follow these steps:

### Step 1: Identify the endpoints of the vector
The endpoints of the vector are given as:
- Point A: \( (4, -4, 0) \)
- Point B: \( (-2, -2, 0) \)

### Step 2: Determine the position vector
The position vector \( \vec{r} \) from point A to point B can be calculated using the formula:
\[ \vec{r} = \vec{r_2} - \vec{r_1} \]
where \( \vec{r_1} = (4, -4, 0) \) and \( \vec{r_2} = (-2, -2, 0) \).

### Step 3: Calculate the components of the vector
Calculating the components:
\[ \vec{r} = (-2 - 4, -2 - (-4), 0 - 0) \]
This simplifies to:
\[ \vec{r} = (-6, 2, 0) \]

### Step 4: Calculate the magnitude of the vector
The magnitude \( |\vec{r}| \) of the vector can be calculated using the formula:
\[ |\vec{r}| = \sqrt{(x^2 + y^2 + z^2)} \]
Substituting the components:
\[ |\vec{r}| = \sqrt{(-6)^2 + (2)^2 + (0)^2} \]
Calculating the squares:
\[ |\vec{r}| = \sqrt{36 + 4 + 0} = \sqrt{40} \]

### Step 5: Simplify the magnitude
We can simplify \( \sqrt{40} \):
\[ |\vec{r}| = \sqrt{4 \times 10} = 2\sqrt{10} \]

### Final Answer
The magnitude of the vector is:
\[ |\vec{r}| = 2\sqrt{10} \]

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