If 𝑢⃗ = <10, 2> and 𝑣⃗ =<-3, 4>, then what is the magnitude of  1/2𝑢⃗ −2𝑣⃗ .

To find the magnitude of the vector \( \frac{1}{2}\mathbf{u} - 2\mathbf{v} \), we will proceed with the following steps:

1. **Calculate \( \frac{1}{2}\mathbf{u} \) and \( -2\mathbf{v} \)**:
   \[
   \frac{1}{2}\mathbf{u} = \frac{1}{2} \times \langle 10, 2 \rangle = \langle 5, 1 \rangle
   \]
   \[
   -2\mathbf{v} = -2 \times \langle -3, 4 \rangle = \langle 6, -8 \rangle
   \]

2. **Find the resultant vector \( \frac{1}{2}\mathbf{u} - 2\mathbf{v} \)**:
   \[
   \frac{1}{2}\mathbf{u} - 2\mathbf{v} = \langle 5, 1 \rangle + \langle 6, -8 \rangle = \langle 11, -7 \rangle
   \]

3. **Calculate the magnitude of the resultant vector**:
   The magnitude of a vector \( \mathbf{w} = \langle a, b \rangle \) is given by \( \|\mathbf{w}\| = \sqrt{a^2 + b^2} \).
   \[
   \|\mathbf{w}\| = \sqrt{11^2 + (-7)^2} = \sqrt{121 + 49} = \sqrt{170}
   \]

4. **Simplify the square root**:
   \[
   \sqrt{170} \approx 13.04
   \]

Thus, the magnitude of \( \frac{1}{2}\mathbf{u} - 2\mathbf{v} \) is approximately **13.04**.

Would you like any further details or have any questions?

Here are some related questions:

1. How can you determine the direction of a vector given its components?
2. What is the geometric interpretation of the magnitude of a vector?
3. How do scalar multiplication and vector addition affect the magnitude of a vector?
4. What is the difference between the cross product and the dot product of vectors?
5. How can you apply vectors to solve real-world problems in physics or engineering?

**Tip:** When working with vectors, breaking down the problem step by step (scaling, adding, and calculating magnitude) can help prevent errors and clarify the process.

Discover how to calculate the magnitude of the vector resulting from the operation 1/2𝑢⃗ −2𝑣⃗. Follow a step-by-step explanation, including scalar multiplication and vector addition. Suitable for advanced high school students and learners of vector mathematics.

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