The expression `||<-8, 4||` represents the magnitude (or length) of the vector \(\mathbf{v} = \langle -8, 4 \rangle\). The magnitude of a vector \(\mathbf{v} = \langle a, b \rangle\) in two-dimensional space is calculated using the formula:

\[
||\mathbf{v}|| = \sqrt{a^2 + b^2}
\]

For the vector \(\langle -8, 4 \rangle\):

\[
||\langle -8, 4 \rangle|| = \sqrt{(-8)^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80}
\]

Simplifying \(\sqrt{80}\):

\[
\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}
\]

So, the magnitude \(||\langle -8, 4 \rangle||\) is \(4\sqrt{5}\).

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the general formula for the magnitude of a vector in three-dimensional space?
2. How does the direction of a vector affect its magnitude?
3. Can you explain the geometric interpretation of a vector's magnitude?
4. How would you normalize the vector \(\langle -8, 4 \rangle\)?
5. How do you find the magnitude of a vector given its initial and terminal points?

**Tip:** To find the magnitude of any vector, square each component, sum them, and then take the square root of the result.

Learn how to calculate the magnitude of the vector (-8, 4) using the formula for vector magnitude in two-dimensional space. This problem involves understanding vector mathematics and applying the formula sqrt(a^2 + b^2) for the vector (-8, 4). Suitable for students in Grades 9-12.

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