If P, Q,R,S are the points `(4,5,3) ,(6,3,4),(2,4,-1)and (0,5,1),` the length of projection RS on PQ is:

Options

A. `4//3`
B. `2//3`
C. `4`
D. 6

Answer

To find the length of the projection of the line segment RS on the line segment PQ, we can follow these steps:

### Step 1: Identify the Points
We have the points:
- \( P(4, 5, 3) \)
- \( Q(6, 3, 4) \)
- \( R(2, 4, -1) \)
- \( S(0, 5, 1) \)

### Step 2: Find the Vectors PQ and RS
The vector \( \overrightarrow{PQ} \) can be calculated as:
\[ \overrightarrow{PQ} = Q - P = (6 - 4, 3 - 5, 4 - 3) = (2, -2, 1) \]

The vector \( \overrightarrow{RS} \) can be calculated as:
\[ \overrightarrow{RS} = S - R = (0 - 2, 5 - 4, 1 - (-1)) = (-2, 1, 2) \]

### Step 3: Calculate the Dot Product of PQ and RS
Now, we compute the dot product \( \overrightarrow{PQ} \cdot \overrightarrow{RS} \):
\[ \overrightarrow{PQ} \cdot \overrightarrow{RS} = (2)(-2) + (-2)(1) + (1)(2) = -4 - 2 + 2 = -4 \]

### Step 4: Find the Magnitude of PQ
Next, we find the magnitude of \( \overrightarrow{PQ} \):
\[ |\overrightarrow{PQ}| = \sqrt{(2)^2 + (-2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

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

### Final Answer
The length of the projection of RS on PQ is \( \frac{4}{3} \).

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