Cosines of angles made by a vector with X,Y axes are `3//5sqrt(2),4//5sqrt(2)` respectively. If the magnitude of the vector is `10sqrt(2)` then that vector is

Options

A. `8hati+6hatj-10hatk`
B. `6hati-8hatj-10hatk`
C. `-6hati-8hatj+10hatk`
D. `6hati+8hatj+10hatk`

Explanation

To solve the problem step by step, let's follow the given information and apply the relevant formulas.

### Step 1: Identify the given values
We are given:
- \( \cos \alpha = \frac{3}{5\sqrt{2}} \)
- \( \cos \beta = \frac{4}{5\sqrt{2}} \)
- Magnitude of the vector \( |\mathbf{r}| = 10\sqrt{2} \)

### Step 2: Find \( \cos \gamma \)
We know that the sum of the squares of the direction cosines equals 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

First, we calculate \( \cos^2 \alpha \) and \( \cos^2 \beta \):
\[ \cos^2 \alpha = \left( \frac{3}{5\sqrt{2}} \right)^2 = \frac{9}{50} \]
\[ \cos^2 \beta = \left( \frac{4}{5\sqrt{2}} \right)^2 = \frac{16}{50} \]

Now, we can find \( \cos^2 \gamma \):
\[ \cos^2 \gamma = 1 - \left( \cos^2 \alpha + \cos^2 \beta \right) = 1 - \left( \frac{9}{50} + \frac{16}{50} \right) = 1 - \frac{25}{50} = \frac{25}{50} = \frac{1}{2} \]

Thus, we have:
\[ \cos \gamma = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 3: Write the vector in terms of its components
The vector \( \mathbf{r} \) can be expressed in terms of its magnitude and direction cosines:
\[ \mathbf{r} = |\mathbf{r}| \left( \cos \alpha \hat{i} + \cos \beta \hat{j} + \cos \gamma \hat{k} \right) \]

Substituting the known values:
\[ \mathbf{r} = 10\sqrt{2} \left( \frac{3}{5\sqrt{2}} \hat{i} + \frac{4}{5\sqrt{2}} \hat{j} + \frac{1}{\sqrt{2}} \hat{k} \right) \]

### Step 4: Simplify the components
Now, we simplify each component:
\[ \mathbf{r} = 10\sqrt{2} \left( \frac{3}{5\sqrt{2}} \hat{i} + \frac{4}{5\sqrt{2}} \hat{j} + \frac{1}{\sqrt{2}} \hat{k} \right) \]

Calculating each term:
1. For \( \hat{i} \):
\[ 10\sqrt{2} \cdot \frac{3}{5\sqrt{2}} = 10 \cdot \frac{3}{5} = 6 \]
2. For \( \hat{j} \):
\[ 10\sqrt{2} \cdot \frac{4}{5\sqrt{2}} = 10 \cdot \frac{4}{5} = 8 \]
3. For \( \hat{k} \):
\[ 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \]

### Step 5: Write the final vector
Putting it all together, we have:
\[ \mathbf{r} = 6\hat{i} + 8\hat{j} + 10\hat{k} \]

### Final Answer
The vector is:
\[ \mathbf{r} = 6\hat{i} + 8\hat{j} + 10\hat{k} \]