To find $4\vec{u} - 2\vec{v}$, we need to multiply each vector by its scalar and then subtract them.

Given:

$\vec{u} = \langle -3, -2 \rangle, \quad \vec{v} = \langle 5, -7 \rangle$

Step 1: Multiply $\vec{u}$ by 4

$4\vec{u} = 4 \times \langle -3, -2 \rangle = \langle 4 \times (-3), 4 \times (-2) \rangle = \langle -12, -8 \rangle$

Step 2: Multiply $\vec{v}$ by 2

$2\vec{v} = 2 \times \langle 5, -7 \rangle = \langle 2 \times 5, 2 \times (-7) \rangle = \langle 10, -14 \rangle$

Step 3: Subtract $2\vec{v}$ from $4\vec{u}$

$4\vec{u} - 2\vec{v} = \langle -12, -8 \rangle - \langle 10, -14 \rangle = \langle -12 - 10, -8 - (-14) \rangle = \langle -22, 6 \rangle$

Thus, $4\vec{u} - 2\vec{v} = \langle -22, 6 \rangle$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you calculate the magnitude of a vector?
2. What is the dot product of $\vec{u}$ and $\vec{v}$?
3. How do you find the angle between two vectors?
4. What is the geometric interpretation of vector subtraction?
5. How do you add two vectors graphically?

Tip: When working with vectors, always be mindful of signs during scalar multiplication and vector operations.