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A unit vector is a vector that has a magnitude of 1. This makes it particularly useful when you want to represent the direction of a vector without considering its length. The process of converting any vector into a unit vector is called normalization. When creating a unit vector, you essentially take the original vector and adjust its length to 1, maintaining the same direction.

• Purpose: Describes direction only.
• Magnitude: Always equal to 1.
• Identification: Notation often involves a caret, like \( \hat{v} \).

For instance, if you have a vector \( \vec{v} = 0.06 \vec{i} - 0.08 \vec{k} \), and you normalize it, you end up with the unit vector \( 0.6 \vec{i} - 0.8 \vec{k} \) as determined in the solution. Its magnitude of 1 is checked with the Pythagorean theorem, ensuring its unit length.

The magnitude of a vector is a measure of its length. It gives you an idea of how long the vector is in its vector space. To find the magnitude of a vector with components in the x, y, and z directions, you use the formula:
\[ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} \]This formula applies the Pythagorean theorem in three dimensions. If a vector has no component in a particular direction (like the y-axis in our original vector example), that component's term becomes zero in the calculation. In the example \( 0.06 \vec{i} - 0.08 \vec{k} \), there is no y-component, so the magnitude is calculated as
\[ |\vec{v}| = \sqrt{0.06^2 + 0 + (-0.08)^2} = \sqrt{0.01} = 0.1 \]This result is important because it sets up the vector for normalization.

Normalization is the process of converting any vector into a unit vector. This involves scaling the vector to have a magnitude of one while keeping its direction unchanged. To normalize a vector, you divide each of its components by its magnitude.
To normalize a vector \( \vec{v} = a \vec{i} + b \vec{j} + c \vec{k} \), follow these steps:

• Calculate the magnitude \( |\vec{v}| \).
• Divide each component by this magnitude:
  \[ \vec{u} = \frac{1}{|\vec{v}|} \vec{v} = \frac{1}{|\vec{v}|} (a \vec{i} + b \vec{j} + c \vec{k}) \]

In the given exercise, the vector \( 0.06 \vec{i} - 0.08 \vec{k} \) was normalized by dividing each component by its magnitude \( 0.1 \), resulting in the unit vector \( 0.6 \vec{i} - 0.8 \vec{k} \). This means that the vector's direction is preserved, but its magnitude is now reduced to 1.