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The image shows a problem involving coordinates and vector operations. Here is the problem along with the steps to solve it:

**Problem:**

Given the coordinates of three points \( A(-1, 4) \), \( B(5, 1) \), and \( C(12, -3) \), the following tasks are required:
1. Find the length of the vector \( \overrightarrow{BC} \).
2. Calculate the length of the projection of vector \( \overrightarrow{BA} \) onto vector \( \overrightarrow{BC} \).

### **Step 1: Find the vector \( \overrightarrow{BC} \)**

To find vector \( \overrightarrow{BC} \), we subtract the coordinates of point \( B \) from those of point \( C \):
\[
\overrightarrow{BC} = \begin{pmatrix} 12 - 5 \\ -3 - 1 \end{pmatrix} = \begin{pmatrix} 7 \\ -4 \end{pmatrix}
\]

### **Step 2: Calculate the length of vector \( \overrightarrow{BC} \)**

The length of vector \( \overrightarrow{BC} \) is given by the formula:
\[
\text{Length of } \overrightarrow{BC} = \sqrt{7^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.06
\]

### **Step 3: Find the vector \( \overrightarrow{BA} \)**

Similarly, we find vector \( \overrightarrow{BA} \) by subtracting the coordinates of \( B \) from \( A \):
\[
\overrightarrow{BA} = \begin{pmatrix} -1 - 5 \\ 4 - 1 \end{pmatrix} = \begin{pmatrix} -6 \\ 3 \end{pmatrix}
\]

### **Step 4: Calculate the projection of vector \( \overrightarrow{BA} \) onto \( \overrightarrow{BC} \)**

The formula for the projection of vector \( \overrightarrow{BA} \) onto \( \overrightarrow{BC} \) is given by:
\[
\text{Proj}_{\overrightarrow{BC}} \overrightarrow{BA} = \frac{\overrightarrow{BA} \cdot \overrightarrow{BC}}{|\overrightarrow{BC}|}
\]

First, find the dot product \( \overrightarrow{BA} \cdot \overrightarrow{BC} \):
\[
\overrightarrow{BA} \cdot \overrightarrow{BC} = (-6)(7) + (3)(-4) = -42 - 12 = -54
\]

Then, divide by the length of \( \overrightarrow{BC} \):
\[
\text{Proj}_{\overrightarrow{BC}} \overrightarrow{BA} = \frac{-54}{\sqrt{65}} \approx -6.68
\]

### **Final Answer:**

- **The length of vector \( \overrightarrow{BC} \) is \( \sqrt{65} \approx 8.06 \).**
- **The length of the projection of vector \( \overrightarrow{BA} \) onto \( \overrightarrow{BC} \) is \( \approx -6.68 \).**

Would you like more details or explanations? 

Here are some related questions you might be interested in:
1. What is the significance of the projection of one vector onto another?
2. How do you interpret the negative sign in the projection length?
3. Can you find the angle between vectors \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \)?
4. How can you determine if two vectors are perpendicular using dot products?
5. What are the real-life applications of vector projections?

**Tip:** Remember that the direction of a vector plays a crucial role in determining the sign of the projection.

Given the coordinates of points A (-1, 4), B (5, 1), and C (12, -3), find the length of the vector BC and the length of the projection of vector BA onto vector BC.

Solve the problem involving vector operations with points A (-1, 4), B (5, 1), and C (12, -3). Learn how to calculate the length of vector BC and the projection of vector BA onto vector BC using essential vector formulas.

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