To find the projection of the line segment \( PQ \) on the plane \( x + y + z = 3 \), we will follow these steps:
### Step 1: Identify Points P and Q
The points given are:
- \( P = (0, 1, 0) \)
- \( Q = (0, 0, 1) \)
### Step 2: Find the Direction Ratios of PQ
The direction ratios of the line segment \( PQ \) can be calculated as follows:
- The direction from \( P \) to \( Q \) is given by:
\[
PQ = Q - P = (0 - 0, 0 - 1, 1 - 0) = (0, -1, 1)
\]
Thus, the direction ratios of \( PQ \) are \( (0, -1, 1) \).
### Step 3: Find the Normal Vector of the Plane
The equation of the plane is given as:
\[
x + y + z = 3
\]
The normal vector \( \mathbf{n} \) to the plane can be derived from the coefficients of \( x, y, z \) in the plane equation:
\[
\mathbf{n} = (1, 1, 1)
\]
### Step 4: Check for Perpendicularity
To check if the line \( PQ \) is perpendicular to the normal of the plane, we calculate the dot product of the direction ratios of \( PQ \) and the normal vector \( \mathbf{n} \):
\[
\mathbf{d} \cdot \mathbf{n} = (0, -1, 1) \cdot (1, 1, 1) = 0 \cdot 1 + (-1) \cdot 1 + 1 \cdot 1 = 0 - 1 + 1 = 0
\]
Since the dot product is zero, \( PQ \) is perpendicular to the normal vector of the plane, indicating that \( PQ \) is parallel to the plane.
### Step 5: Calculate the Length of PQ
The length of the line segment \( PQ \) can be calculated using the distance formula:
\[
\text{Length of } PQ = \sqrt{(0 - 0)^2 + (1 - 0)^2 + (0 - 1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2}
\]
### Step 6: Conclusion
Since \( PQ \) is parallel to the plane, the projection of \( PQ \) on the plane is equal to the length of \( PQ \). Therefore, the projection of \( PQ \) on the plane \( x + y + z = 3 \) is:
\[
\text{Projection of } PQ = \sqrt{2}
\]
### Final Answer
The projection of \( PQ \) on the plane \( x + y + z = 3 \) is \( \sqrt{2} \).
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