Certainly! Let's break this problem down step by step to find the unit vector in the direction of [tex]\( -2\overrightarrow{v} \)[/tex].
### Step 1: Understand the Vector [tex]\( -2\overrightarrow{v} \)[/tex]
The vector [tex]\( -2\overrightarrow{v} \)[/tex] is simply a vector that is scaled by -2 in the direction of [tex]\( \overrightarrow{v} \)[/tex]. For simplicity, let's assume that [tex]\( \overrightarrow{v} = (1, 0) \)[/tex].
Hence,
[tex]\[ -2\overrightarrow{v} = -2 \cdot (1, 0) = (-2, 0) \][/tex]
### Step 2: Calculate the Magnitude of [tex]\( -2\overrightarrow{v} \)[/tex]
The magnitude of a vector [tex]\((x, y)\)[/tex] is given by:
[tex]\[ \| \overrightarrow{v} \| = \sqrt{x^2 + y^2} \][/tex]
For the vector [tex]\( (-2, 0) \)[/tex]:
[tex]\[ \| -2\overrightarrow{v} \| = \sqrt{(-2)^2 + 0^2} = \sqrt{4} = 2.0 \][/tex]
### Step 3: Find the Unit Vector
A unit vector in the direction of a given vector [tex]\( \overrightarrow{v} \)[/tex] is obtained by dividing each component of the vector by its magnitude.
For the vector [tex]\( (-2, 0) \)[/tex], whose magnitude we just calculated as 2.0, the unit vector [tex]\( \overrightarrow{u} \)[/tex] is:
[tex]\[ \overrightarrow{u} = \left( \frac{-2}{2.0}, \frac{0}{2.0} \right) \][/tex]
[tex]\[ \overrightarrow{u} = (-1.0, 0.0) \][/tex]
### Conclusion
Thus, the unit vector in the direction of [tex]\( -2\overrightarrow{v} \)[/tex] is [tex]\( (-1.0, 0.0) \)[/tex] and the magnitude of the original vector [tex]\( -2\overrightarrow{v} \)[/tex] is 2.0.
