(a) Let P = (2, 0, 1), Q = (3, 1, 0) and R = (4, 3, 5). Calculate the projection of the vector
−→
P R onto the vector −→P Q

Let's calculate the projection of vector \( \overrightarrow{PR} \) onto vector \( \overrightarrow{PQ} \).

Given points:
- \( P = (2, 0, 1) \)
- \( Q = (3, 1, 0) \)
- \( R = (4, 3, 5) \)

### Step 1: Find vectors \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \).

#### Vector \( \overrightarrow{PQ} \):
\[
\overrightarrow{PQ} = Q - P = (3, 1, 0) - (2, 0, 1) = (3 - 2, 1 - 0, 0 - 1) = (1, 1, -1)
\]

#### Vector \( \overrightarrow{PR} \):
\[
\overrightarrow{PR} = R - P = (4, 3, 5) - (2, 0, 1) = (4 - 2, 3 - 0, 5 - 1) = (2, 3, 4)
\]

### Step 2: Calculate the projection of \( \overrightarrow{PR} \) onto \( \overrightarrow{PQ} \).

The formula for the projection of vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is:
\[
\text{Proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}
\]
Here, \( \mathbf{a} = \overrightarrow{PR} \) and \( \mathbf{b} = \overrightarrow{PQ} \).

#### Step 3: Compute the dot products.
\[
\overrightarrow{PR} \cdot \overrightarrow{PQ} = (2, 3, 4) \cdot (1, 1, -1) = (2 \times 1) + (3 \times 1) + (4 \times -1) = 2 + 3 - 4 = 1
\]
\[
\overrightarrow{PQ} \cdot \overrightarrow{PQ} = (1, 1, -1) \cdot (1, 1, -1) = (1 \times 1) + (1 \times 1) + (-1 \times -1) = 1 + 1 + 1 = 3
\]

#### Step 4: Calculate the projection.
\[
\text{Proj}_{\overrightarrow{PQ}} \overrightarrow{PR} = \frac{1}{3} \overrightarrow{PQ} = \frac{1}{3} (1, 1, -1)
\]
\[
\text{Proj}_{\overrightarrow{PQ}} \overrightarrow{PR} = \left( \frac{1}{3}, \frac{1}{3}, -\frac{1}{3} \right)
\]

Thus, the projection of \( \overrightarrow{PR} \) onto \( \overrightarrow{PQ} \) is \( \left( \frac{1}{3}, \frac{1}{3}, -\frac{1}{3} \right) \).

Would you like further clarification or details on any step?

Here are five related questions:
1. How do you calculate the angle between two vectors?
2. What is the geometric interpretation of a vector projection?
3. How can you find the magnitude of a projected vector?
4. How does the dot product relate to the cosine of the angle between two vectors?
5. What are the conditions for two vectors to be orthogonal?

**Tip:** Always double-check vector subtraction when calculating the initial direction vectors to avoid errors.

Let P = (2, 0, 1), Q = (3, 1, 0) and R = (4, 3, 5). Calculate the projection of the vector PR onto the vector PQ.

Learn how to calculate the projection of vector PR onto vector PQ given the points P(2, 0, 1), Q(3, 1, 0), and R(4, 3, 5). This problem involves vector algebra, vector projections, and dot products, making it suitable for students in Grades 10-12.

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