find the magnitude of the vector. write your answer in simplified form.

1. First, assume the vector has initial - point at the origin ((0,0)) and terminal - point ((8, - 4)).
   • The magnitude of a two - dimensional vector (\vec{v}=(x,y)) is given by the formula (|\vec{v}|=\sqrt{x^{2}+y^{2}}).
   • Here, (x = 8) and (y=-4).
2. Then, substitute the values of (x) and (y) into the formula:
   • (|\vec{v}|=\sqrt{8^{2}+(-4)^{2}}).
   • Calculate (8^{2}=64) and ((-4)^{2}=16). So, (|\vec{v}|=\sqrt{64 + 16}).
   • Add the values inside the square root: (64+16 = 80). Then (|\vec{v}|=\sqrt{80}).
3. Next, simplify the square root:
   • Factor 80: (80=16\times5).
   • Using the property (\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}) ((a = 16), (b = 5) and (a\geq0), (b\geq0)), we have (|\vec{v}|=\sqrt{16}\cdot\sqrt{5}).
   • Since (\sqrt{16}=4), then (|\vec{v}| = 4\sqrt{5}).

Explanation:

Step1: Identify vector components

Let vector (\vec{v}=(8,-4))

Step2: Apply magnitude formula

(|\vec{v}|=\sqrt{8^{2}+(-4)^{2}}=\sqrt{64 + 16}=\sqrt{80})

Step3: Simplify square - root

(|\vec{v}|=\sqrt{16\times5}=4\sqrt{5})

Answer:

(4\sqrt{5})