Given that a=⟨−2,5⟩
 and b=⟨4,1⟩
, draw a graphical representation of 2a−3b
. Then find the resultant vector algebraically to verify your resultant.

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?

---

### 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.

Given that a=⟨−2,5⟩ and b=⟨4,1⟩, draw a graphical representation of 2a−3b. Then find the resultant vector algebraically to verify your resultant.

Explore how to calculate the resultant vector from 2a - 3b using vector algebra. This problem covers vector operations and graphical representation of vectors, suitable for Grades 9-12.

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