The given problem asks us to find a vector that is oppositely directed to $\mathbf{v} = \langle 3, -4 \rangle$ and is half the length of $\mathbf{v}$.

Step 1: Determine the opposite direction

To get a vector that is oppositely directed to $\mathbf{v}$, we multiply $\mathbf{v}$ by $-1$. This will reverse its direction.

$\mathbf{-v} = -1 \times \langle 3, -4 \rangle = \langle -3, 4 \rangle$

Step 2: Find half the length of the vector

Next, we are asked to find a vector that is half the length of $\mathbf{v}$. To do this, we scale the vector $\mathbf{-v}$ by $\frac{1}{2}$.

$\text{Half of } \mathbf{-v} = \frac{1}{2} \times \langle -3, 4 \rangle = \langle -\frac{3}{2}, 2 \rangle$

Thus, the vector that is oppositely directed to $\mathbf{v}$ and half its length is:

$\boxed{\langle -\frac{3}{2}, 2 \rangle}$

Would you like more details on any of the steps?

Related Questions:

1. What is the magnitude (length) of the vector $\mathbf{v} = \langle 3, -4 \rangle$?
2. How would you find a vector that is twice as long as $\mathbf{v}$ but in the same direction?
3. What is the unit vector in the direction of $\mathbf{v}$?
4. How do you calculate the dot product of two vectors?
5. How do vectors change when they are rotated in a plane?

Tip:

To reverse a vector's direction, multiplying it by $-1$ is the quickest way!