To find a unit vector that has the same direction as the given vector [tex]\( v = (5, -4) \)[/tex], we need to follow these steps:
1. Calculate the magnitude of the vector [tex]\( v \)[/tex]:
The magnitude of a vector [tex]\( v = (x, y) \)[/tex] is given by:
[tex]\[
\|v\| = \sqrt{x^2 + y^2}
\][/tex]
For the vector [tex]\( v = (5, -4) \)[/tex]:
[tex]\[
\|v\| = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41}
\][/tex]
2. Determine the unit vector [tex]\(\hat{v}\)[/tex] in the direction of [tex]\( v \)[/tex]:
A unit vector in the direction of [tex]\( v \)[/tex] is given by:
[tex]\[
\hat{v} = \left( \frac{x}{\|v\|}, \frac{y}{\|v\|} \right)
\][/tex]
Substituting [tex]\( x = 5 \)[/tex], [tex]\( y = -4 \)[/tex], and [tex]\( \|v\| = \sqrt{41} \)[/tex]:
[tex]\[
\hat{v} = \left( \frac{5}{\sqrt{41}}, \frac{-4}{\sqrt{41}} \right)
\][/tex]
So, the unit vector that has the same direction as the given vector [tex]\( v = (5, -4) \)[/tex] is:
[tex]\[
\left( \frac{5}{\sqrt{41}}, \frac{-4}{\sqrt{41}} \right)
\][/tex]
Hence, a unit vector in the same direction as [tex]\( v \)[/tex] is:
[tex]\[
\left( \frac{5}{\sqrt{41}}, \frac{-4}{\sqrt{41}} \right)
\][/tex]
