To solve this, we’ll go through two main steps: (1) calculating the vector $2\mathbf{a} - 3\mathbf{b}$ algebraically, and (2) graphically representing this vector. Let's begin.

Step 1: Calculate $2\mathbf{a} - 3\mathbf{b}$ Algebraically

Given: $\mathbf{a} = \langle -2, 5 \rangle$ $\mathbf{b} = \langle 4, 1 \rangle$

1. Multiply $\mathbf{a}$ by 2: $2\mathbf{a} = 2 \cdot \langle -2, 5 \rangle = \langle -4, 10 \rangle$

2. Multiply $\mathbf{b}$ by 3: $3\mathbf{b} = 3 \cdot \langle 4, 1 \rangle = \langle 12, 3 \rangle$

3. Calculate $2\mathbf{a} - 3\mathbf{b}$: $2\mathbf{a} - 3\mathbf{b} = \langle -4, 10 \rangle - \langle 12, 3 \rangle$ $= \langle -4 - 12, 10 - 3 \rangle$ $= \langle -16, 7 \rangle$

So, the resultant vector is: $2\mathbf{a} - 3\mathbf{b} = \langle -16, 7 \rangle$

Step 2: Graphical Representation

1. Draw vectors $\mathbf{a}$ and $\mathbf{b}$: Represent $\mathbf{a} = \langle -2, 5 \rangle$ and $\mathbf{b} = \langle 4, 1 \rangle$ on a coordinate system.

2. Scale $\mathbf{a}$ and $\mathbf{b}$ appropriately: Draw $2\mathbf{a}$ by extending $\mathbf{a}$ to twice its length, and $3\mathbf{b}$ by extending $\mathbf{b}$ to three times its length.

3. Subtract the scaled vectors: To get $2\mathbf{a} - 3\mathbf{b}$, place the vector $3\mathbf{b}$ with its tail at the head of $2\mathbf{a}$, but in the opposite direction, since we are subtracting $3\mathbf{b}$.

4. Draw the resultant vector $\langle -16, 7 \rangle$: The vector $\langle -16, 7 \rangle$ can now be represented on the graph, originating from the origin to show the resultant direction and magnitude.

Would you like a graphical illustration, or do you have any questions about these steps?

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Additional Questions to Consider

1. What are the magnitudes of vectors $\mathbf{a}$, $\mathbf{b}$, and $2\mathbf{a} - 3\mathbf{b}$?
2. How would the direction of the resultant vector change if the sign in $2\mathbf{a} - 3\mathbf{b}$ was reversed to $2\mathbf{a} + 3\mathbf{b}$?
3. How does vector scaling affect the angle of the resultant vector relative to the x-axis?
4. What other applications could benefit from calculating vector differences?
5. How could this vector result be visualized in three dimensions if a third component were added?

Tip:

Always check each component step-by-step to avoid calculation errors when adding or subtracting vectors algebraically.