If two points are `P(7, -5, 11) and Q(-2, 8, 13)`, then the projection of PQ on a straight line with direction cosines `(1)/(2), (2)/(3),(2)/(3)` is

Options

A. `(1)/(2)`
B. `(26)/(3)`
C. `(4)/(3)`
D. 6

Answer

To find the projection of the line segment joining the points \( P(7, -5, 11) \) and \( Q(-2, 8, 13) \) onto a straight line with direction cosines \( \left( \frac{1}{2}, \frac{2}{3}, \frac{2}{3} \right) \), we can follow these steps:

### Step 1: Find the coordinates of points P and Q
The coordinates of point \( P \) are \( (7, -5, 11) \) and the coordinates of point \( Q \) are \( (-2, 8, 13) \).

### Step 2: Calculate the vector \( \overrightarrow{PQ} \)
The vector \( \overrightarrow{PQ} \) can be calculated using the formula:
\[ \overrightarrow{PQ} = Q - P = (-2 - 7, 8 - (-5), 13 - 11) \]
Calculating each component:
\[ \overrightarrow{PQ} = (-9, 13, 2) \]

### Step 3: Identify the direction cosines
The direction cosines of the line are given as:
\[ l = \frac{1}{2}, \quad m = \frac{2}{3}, \quad n = \frac{2}{3} \]

### Step 4: Use the projection formula
The projection of the vector \( \overrightarrow{PQ} \) onto the line with direction cosines \( (l, m, n) \) is given by the formula:
\[ \text{Projection} = \overrightarrow{PQ} \cdot (l, m, n) \]
Calculating the dot product:
\[ \text{Projection} = (-9) \cdot \frac{1}{2} + (13) \cdot \frac{2}{3} + (2) \cdot \frac{2}{3} \]

### Step 5: Calculate each term
Calculating each term separately:
1. \( -9 \cdot \frac{1}{2} = -\frac{9}{2} \)
2. \( 13 \cdot \frac{2}{3} = \frac{26}{3} \)
3. \( 2 \cdot \frac{2}{3} = \frac{4}{3} \)

### Step 6: Combine the results
Now, we combine these results:
\[ \text{Projection} = -\frac{9}{2} + \frac{26}{3} + \frac{4}{3} \]

### Step 7: Find a common denominator
The common denominator for \( 2 \) and \( 3 \) is \( 6 \):
- Convert \( -\frac{9}{2} \) to sixths:
\[ -\frac{9}{2} = -\frac{27}{6} \]
- Convert \( \frac{26}{3} \) to sixths:
\[ \frac{26}{3} = \frac{52}{6} \]
- Convert \( \frac{4}{3} \) to sixths:
\[ \frac{4}{3} = \frac{8}{6} \]

### Step 8: Combine the fractions
Now we can combine:
\[ \text{Projection} = -\frac{27}{6} + \frac{52}{6} + \frac{8}{6} = \frac{-27 + 52 + 8}{6} = \frac{33}{6} = \frac{11}{2} \]

### Final Answer
Thus, the projection of \( PQ \) on the given line is:
\[ \frac{11}{2} \]