The direction cosines of the vector `3hati-4hatj+5hatk` are

Options

A. `(3)/(5),(-4)/(5),(1)/(5)`
B. `(3)/(5sqrt(2)),(-4)/(5sqrt(2)),(1)/(sqrt(2))`
C. `(3)/(sqrt(2)),(-4)/(sqrt(2)),(1)/(sqrt(2))`
D. `(3)/(5sqrt(2)),(4)/(5sqrt(2)),(1)/(sqrt(2))`.

Explanation

To find the direction cosines of the vector \( \mathbf{v} = 3\hat{i} - 4\hat{j} + 5\hat{k} \), we will follow these steps:

### Step 1: Calculate the magnitude of the vector
The magnitude \( |\mathbf{v}| \) of the vector \( \mathbf{v} = 3\hat{i} - 4\hat{j} + 5\hat{k} \) is calculated using the formula:
\[ |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \]
where \( x, y, z \) are the components of the vector.

For our vector:
- \( x = 3 \)
- \( y = -4 \)
- \( z = 5 \)

Thus,
\[ |\mathbf{v}| = \sqrt{3^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \]

### Step 2: Calculate the direction cosines
The direction cosines \( l, m, n \) of the vector are given by:
\[ l = \frac{x}{|\mathbf{v}|}, \quad m = \frac{y}{|\mathbf{v}|}, \quad n = \frac{z}{|\mathbf{v}|} \]

Substituting the values we have:
\[ l = \frac{3}{5\sqrt{2}}, \quad m = \frac{-4}{5\sqrt{2}}, \quad n = \frac{5}{5\sqrt{2}} \]

### Step 3: Simplify the direction cosines
Now we simplify the direction cosines:
\[ l = \frac{3}{5\sqrt{2}}, \quad m = \frac{-4}{5\sqrt{2}}, \quad n = \frac{1}{\sqrt{2}} \]

Thus, the direction cosines of the vector \( 3\hat{i} - 4\hat{j} + 5\hat{k} \) are:
\[ \left( \frac{3}{5\sqrt{2}}, \frac{-4}{5\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]

### Final Answer
The direction cosines of the vector \( 3\hat{i} - 4\hat{j} + 5\hat{k} \) are:
\[ \left( \frac{3}{5\sqrt{2}}, \frac{-4}{5\sqrt{2}}, \frac{1}{\sqrt{2}} \right) \]

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