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A unit vector is essentially a direction indicator and has a magnitude of 1. It is derived by scaling a given vector so that its magnitude becomes one, keeping its direction unchanged.
To find a unit vector, you simply divide each component of the original vector by the vector's magnitude. This means the unit vector points in the same direction as the original, but its length is precisely one unit.
Using the problem, we start with the vector \( \mathbf{v} = 12\mathbf{i} - 5\mathbf{k} \). Its magnitude is found as 13. Thus, the unit vector \( \mathbf{u} \) in this direction is:

• \( \mathbf{u} = \frac{12}{13}\mathbf{i} + \frac{-5}{13}\mathbf{k} \)

This shows that unit vectors help in extracting the pure direction of a vector, making them important in many mathematical and physics problems.

Vector scaling involves changing the length of a vector while preserving its direction. This is done by multiplying the vector by a scalar, a real number that indicates how much to stretch or shrink the vector.
In our example, we scale the unit vector \( \mathbf{u} \), which has a magnitude of 1, to reach a new calculated vector with a magnitude of 7, as desired.
By multiplying each component of \( \mathbf{u} \) by 7, we achieve this new vector:

• \( 7 \times \left( \frac{12}{13}\mathbf{i} + \frac{-5}{13}\mathbf{k} \right) = \frac{84}{13}\mathbf{i} + \frac{-35}{13}\mathbf{k} \)

The resulting vector retains the direction of the original vector \( \mathbf{v} \), but now it stretches to the desired length of 7. This procedure is key in various fields like physics and engineering when translating forces or velocities to desired scales.

The magnitude of a vector provides the length of the vector in space. It is calculated using the formula \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the vector's components.
The magnitude represents how long a vector is, ignoring its direction. For a vector \( \mathbf{v} = 12\mathbf{i} - 5\mathbf{k} \), the magnitude is calculated as follows:

• \( \|\mathbf{v}\| = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \)

Understanding how to calculate vector magnitude is critical for performing operations like normalization, comparing vectors, and applying vectors practically in situations requiring length measurement such as in navigation or computer graphics.