https://file.aiquickdraw.com/mathbot/file/17295313785158a5zows7.jpg

To solve this problem, we need to find the vector \( \mathbf{a} - 2\mathbf{b} \), given the column vectors:

\[
\mathbf{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}
\]

### Step 1: Calculate \( 2\mathbf{b} \)

\[
2\mathbf{b} = 2 \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}
\]

### Step 2: Subtract \( 2\mathbf{b} \) from \( \mathbf{a} \)

\[
\mathbf{a} - 2\mathbf{b} = \begin{pmatrix} 5 \\ 2 \end{pmatrix} - \begin{pmatrix} 6 \\ -2 \end{pmatrix} = \begin{pmatrix} 5 - 6 \\ 2 - (-2) \end{pmatrix} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}
\]

### Step 3: Plot the vector on the grid

Starting from the given point (5, 2) on the grid, we move 1 unit to the left (negative x-direction) and 4 units up (positive y-direction). This places the endpoint of the vector at the coordinates (4, 6).

The vector \( \mathbf{a} - 2\mathbf{b} \) is drawn from the starting point (5, 2) to the point (4, 6). 

Would you like further details or have any questions?

### Related Questions:
1. How would you calculate \( \mathbf{a} + \mathbf{b} \)?
2. What is the magnitude of vector \( \mathbf{a} - 2\mathbf{b} \)?
3. How would you find the direction of \( \mathbf{a} - 2\mathbf{b} \) in degrees?
4. Can you express the vector \( \mathbf{b} \) in terms of its magnitude and direction?
5. What happens if we multiply \( \mathbf{a} \) by a scalar?

### Tip:
When working with vectors, break the problem down into scalar multiplications and simple addition or subtraction for each component.

Here are two column vectors: a = (5, 2), b = (3, -1). On the grid below, draw the vector a - 2b starting from the dot.

Learn how to draw the vector a - 2b on a grid, given a = (5, 2) and b = (3, -1). This problem involves vector subtraction and scalar multiplication, providing a detailed step-by-step solution for students in Grades 9-11.

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