Find magnitude and direction cosines of the vector, `A= (3hati - 4hatj+5hatk).`

Explanation

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

### Step 1: Calculate the Magnitude of the Vector

The magnitude \( |\mathbf{A}| \) of the vector \( \mathbf{A} \) is given by the formula:

\[ |\mathbf{A}| = \sqrt{(A_x)^2 + (A_y)^2 + (A_z)^2} \]

Where \( A_x, A_y, A_z \) are the components of the vector along the \( \hat{i}, \hat{j}, \hat{k} \) directions respectively.

For our vector \( \mathbf{A} = 3\hat{i} - 4\hat{j} + 5\hat{k} \):

- \( A_x = 3 \)
- \( A_y = -4 \)
- \( A_z = 5 \)

Now substituting these values into the formula:

\[ |\mathbf{A}| = \sqrt{(3)^2 + (-4)^2 + (5)^2} \]

Calculating the squares:

\[ |\mathbf{A}| = \sqrt{9 + 16 + 25} \]

Now summing these values:

\[ |\mathbf{A}| = \sqrt{50} \]

This can be simplified to:

\[ |\mathbf{A}| = 5\sqrt{2} \]

### Step 2: Calculate the Direction Cosines

The direction cosines \( \cos \alpha, \cos \beta, \cos \gamma \) are given by the formulas:

\[ \cos \alpha = \frac{A_x}{|\mathbf{A}|}, \quad \cos \beta = \frac{A_y}{|\mathbf{A}|}, \quad \cos \gamma = \frac{A_z}{|\mathbf{A}|} \]

Substituting the values we have:

1. **For \( \cos \alpha \)**:

\[ \cos \alpha = \frac{3}{5\sqrt{2}} \]

2. **For \( \cos \beta \)**:

\[ \cos \beta = \frac{-4}{5\sqrt{2}} \]

3. **For \( \cos \gamma \)**:

\[ \cos \gamma = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Final Results

- The magnitude of the vector \( \mathbf{A} \) is \( 5\sqrt{2} \).
- The direction cosines are:
- \( \cos \alpha = \frac{3}{5\sqrt{2}} \)
- \( \cos \beta = \frac{-4}{5\sqrt{2}} \)
- \( \cos \gamma = \frac{1}{\sqrt{2}} \)

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