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.