Find the magnitude of the vector :
`hat(i)-3hat(j)+4hat(k)`.

Answer

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

### Step 1: Identify the components of the vector
The vector \( \mathbf{A} \) can be expressed in terms of its components:
- The coefficient of \( \hat{i} \) is \( 1 \).
- The coefficient of \( \hat{j} \) is \( -3 \).
- The coefficient of \( \hat{k} \) is \( 4 \).

### Step 2: Write the formula for the magnitude of the vector
The magnitude of a vector \( \mathbf{A} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by the formula:
\[ |\mathbf{A}| = \sqrt{a^2 + b^2 + c^2} \]
where \( a \), \( b \), and \( c \) are the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) respectively.

### Step 3: Substitute the values into the formula
Substituting the values we identified:
- \( a = 1 \)
- \( b = -3 \)
- \( c = 4 \)

The formula becomes:
\[ |\mathbf{A}| = \sqrt{1^2 + (-3)^2 + 4^2} \]

### Step 4: Calculate the squares of the components
Now, calculate the squares:
- \( 1^2 = 1 \)
- \( (-3)^2 = 9 \)
- \( 4^2 = 16 \)

### Step 5: Add the squares together
Now, add these values together:
\[ 1 + 9 + 16 = 26 \]

### Step 6: Take the square root
Finally, take the square root of the sum:
\[ |\mathbf{A}| = \sqrt{26} \]

### Conclusion
The magnitude of the vector \( \mathbf{A} = \hat{i} - 3\hat{j} + 4\hat{k} \) is:
\[ |\mathbf{A}| = \sqrt{26} \]

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