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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!

Find the vector that is oppositely directed to v = <3, -4> and half the length of v.

Learn how to find a vector that is oppositely directed to v = <3, -4> and is half its length. This problem involves vector operations and scaling, suitable for students in Grades 9-12.

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