The projection of the line joining the ponts `(3,4,5)` and `(4,6,3)` on the line joining the points (-1,2,4) and (1,0,5) is

Options

A. `4/3`
B. `2/3`
C. `1/3`
D. `1/2`

Explanation

To find the projection of the line joining the points \( A(3, 4, 5) \) and \( B(4, 6, 3) \) on the line joining the points \( C(-1, 2, 4) \) and \( D(1, 0, 5) \), we can follow these steps:

### Step 1: Find the vector \( \overrightarrow{AB} \)
The vector \( \overrightarrow{AB} \) can be calculated as:
\[ \overrightarrow{AB} = B - A = (4 - 3) \hat{i} + (6 - 4) \hat{j} + (3 - 5) \hat{k} \]
Calculating this gives:
\[ \overrightarrow{AB} = 1 \hat{i} + 2 \hat{j} - 2 \hat{k} = \hat{i} + 2\hat{j} - 2\hat{k} \]

### Step 2: Find the vector \( \overrightarrow{CD} \)
The vector \( \overrightarrow{CD} \) can be calculated as:
\[ \overrightarrow{CD} = D - C = (1 - (-1)) \hat{i} + (0 - 2) \hat{j} + (5 - 4) \hat{k} \]
Calculating this gives:
\[ \overrightarrow{CD} = 2 \hat{i} - 2 \hat{j} + 1 \hat{k} = 2\hat{i} - 2\hat{j} + \hat{k} \]

### Step 3: Calculate the dot product \( \overrightarrow{AB} \cdot \overrightarrow{CD} \)
Now, we compute the dot product:
\[ \overrightarrow{AB} \cdot \overrightarrow{CD} = (1)(2) + (2)(-2) + (-2)(1) \]
Calculating this gives:
\[ \overrightarrow{AB} \cdot \overrightarrow{CD} = 2 - 4 - 2 = -4 \]

### Step 4: Find the magnitude of \( \overrightarrow{CD} \)
The magnitude of \( \overrightarrow{CD} \) is calculated as:
\[ |\overrightarrow{CD}| = \sqrt{(2)^2 + (-2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 5: Calculate the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{CD} \)
The projection of \( \overrightarrow{AB} \) on \( \overrightarrow{CD} \) is given by the formula:
\[ \text{Projection} = \frac{\overrightarrow{AB} \cdot \overrightarrow{CD}}{|\overrightarrow{CD}|} \]
Substituting the values we found:
\[ \text{Projection} = \frac{-4}{3} \]

### Step 6: Find the magnitude of the projection
The magnitude of the projection is:
\[ |\text{Projection}| = \left| \frac{-4}{3} \right| = \frac{4}{3} \]

### Final Result
Thus, the projection of the line joining the points \( (3, 4, 5) \) and \( (4, 6, 3) \) on the line joining the points \( (-1, 2, 4) \) and \( (1, 0, 5) \) is \( \frac{4}{3} \).

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