The projection of the line segment joining the points `A(-1,0,3)` and `B(2,5,1)` on the line whose direction ratios are proportional to 6,2,3 is

Options

A. `10/7`
B. `22/7`
C. `18/7`
D. none of these

Explanation

To find the projection of the line segment joining the points \( A(-1,0,3) \) and \( B(2,5,1) \) on the line whose direction ratios are proportional to \( 6, 2, 3 \), we can follow these steps:

### Step 1: Find the vector \( \overrightarrow{AB} \)

The vector \( \overrightarrow{AB} \) can be calculated by subtracting the coordinates of point \( A \) from point \( B \):

\[ \overrightarrow{AB} = B - A = (2 - (-1), 5 - 0, 1 - 3) = (2 + 1, 5, 1 - 3) = (3, 5, -2) \]

### Step 2: Identify the direction vector \( \overrightarrow{D} \)

The direction ratios given are proportional to \( 6, 2, 3 \). Therefore, we can take the direction vector \( \overrightarrow{D} \) as:

\[ \overrightarrow{D} = (6, 2, 3) \]

### Step 3: Calculate the dot product \( \overrightarrow{AB} \cdot \overrightarrow{D} \)

The dot product of the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{D} \) is calculated as follows:

\[ \overrightarrow{AB} \cdot \overrightarrow{D} = (3)(6) + (5)(2) + (-2)(3) \]

Calculating each term:

\[ = 18 + 10 - 6 = 22 \]

### Step 4: Calculate the magnitude of the direction vector \( \overrightarrow{D} \)

The magnitude of the vector \( \overrightarrow{D} \) is given by:

\[ |\overrightarrow{D}| = \sqrt{6^2 + 2^2 + 3^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

### Step 5: Calculate the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{D} \)

The projection of \( \overrightarrow{AB} \) on \( \overrightarrow{D} \) is given by the formula:

\[ \text{Projection} = \frac{\overrightarrow{AB} \cdot \overrightarrow{D}}{|\overrightarrow{D}|} \]

Substituting the values we calculated:

\[ \text{Projection} = \frac{22}{7} \]

### Final Answer

Thus, the projection of the line segment joining points \( A \) and \( B \) on the line whose direction ratios are proportional to \( 6, 2, 3 \) is:

\[ \frac{22}{7} \]

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