In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 6i

Verified step by step guidance

1. Identify the given vector \( \mathbf{v} = 6\mathbf{i} \). This means the vector has components \( (6, 0) \) in the 2D plane, where \( \mathbf{i} \) is the unit vector along the x-axis.
2. Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula for the magnitude of a vector: \[ \\| \mathbf{v} \\| = \\sqrt{(6)^2 + (0)^2} \]
3. Simplify the magnitude expression to find the length of \( \mathbf{v} \). This will give you a scalar value representing how long the vector is.
4. To find the unit vector in the same direction as \( \mathbf{v} \), divide each component of \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \frac{1}{\| \mathbf{v} \\|} \mathbf{v} = \left( \frac{6}{\| \mathbf{v} \\|}, \frac{0}{\| \mathbf{v} \\|} \right) \]
5. Express the unit vector \( \mathbf{u} \) clearly, showing that it has the same direction as \( \mathbf{v} \) but a magnitude of 1.

Key Concepts

Vector Direction

The direction of a vector is the orientation it points in space, independent of its magnitude. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction helps in finding unit vectors that preserve this orientation.

Magnitude (Norm) of a Vector

The magnitude of a vector is its length, calculated as the square root of the sum of the squares of its components. For a vector v = 6i, the magnitude is |v| = 6. This value is essential for normalizing the vector to find a unit vector.

Unit Vector

A unit vector has a magnitude of 1 and indicates direction only. To find a unit vector in the same direction as v, divide v by its magnitude. This process normalizes the vector, producing a vector with length one but the same direction.