find the magnitude of the resultant vector.\nround to the nearest hundredth.

Explanation:

Step1: Find the resultant vector components

To find the resultant vector (\vec{R}) of (\vec{v} = (6, 8)) and (\vec{w} = (-4, 12)), we add their corresponding components. The (x)-component of (\vec{R}) is (6 + (-4)=6 - 4 = 2). The (y)-component of (\vec{R}) is (8+12 = 20). So, (\vec{R}=(2,20)).

Step2: Calculate the magnitude of (\vec{R})

The magnitude of a vector ((x,y)) is given by the formula (\vert\vec{R}\vert=\sqrt{x^{2}+y^{2}}). For (\vec{R}=(2,20)), we substitute (x = 2) and (y = 20) into the formula: (\vert\vec{R}\vert=\sqrt{2^{2}+20^{2}}=\sqrt{4 + 400}=\sqrt{404})

Step3: Compute the numerical value and round

We calculate the value of (\sqrt{404}\approx20.09975124). Rounding to the nearest hundredth (two decimal places), we get (20.10).

Answer:

(20.10)