Find the direction ratios and the direction cosines of the vector joining the points `A(2,1,-2)` and `B(3,5,-4)`.

Answer

To find the direction ratios and the direction cosines of the vector joining the points \( A(2, 1, -2) \) and \( B(3, 5, -4) \), we can follow these steps:

### Step 1: Find the vector \( \vec{AB} \)
The vector \( \vec{AB} \) can be found by subtracting the coordinates of point \( A \) from the coordinates of point \( B \).

\[ \vec{AB} = B - A = (3, 5, -4) - (2, 1, -2) \]

Calculating this gives:

\[ \vec{AB} = (3 - 2, 5 - 1, -4 + 2) = (1, 4, -2) \]

### Step 2: Identify the direction ratios
The direction ratios of the vector \( \vec{AB} \) are simply the components of the vector itself.

Thus, the direction ratios \( (A, B, C) \) are:

\[ (1, 4, -2) \]

### Step 3: Calculate the magnitude of the vector \( \vec{AB} \)
The magnitude of the vector \( \vec{AB} \) is given by the formula:

\[ |\vec{AB}| = \sqrt{(1)^2 + (4)^2 + (-2)^2} \]

Calculating this gives:

\[ |\vec{AB}| = \sqrt{1 + 16 + 4} = \sqrt{21} \]

### Step 4: Find the direction cosines
The direction cosines are found using the formula:

\[ \cos \alpha = \frac{A}{|\vec{AB}|}, \quad \cos \beta = \frac{B}{|\vec{AB}|}, \quad \cos \gamma = \frac{C}{|\vec{AB}|} \]

Substituting the values we found:

\[ \cos \alpha = \frac{1}{\sqrt{21}}, \quad \cos \beta = \frac{4}{\sqrt{21}}, \quad \cos \gamma = \frac{-2}{\sqrt{21}} \]

### Final Result
The direction ratios are \( (1, 4, -2) \) and the direction cosines are \( \left( \frac{1}{\sqrt{21}}, \frac{4}{\sqrt{21}}, \frac{-2}{\sqrt{21}} \right) \).

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