Answer
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Work Step by Step
The vector projection of a vector \( \mathbf{u} \) onto a vector \( \mathbf{v} \) is a vector that represents the component of \( \mathbf{u} \) that lies in the direction of \( \mathbf{v} \). It is denoted as \( \text{proj}_{\mathbf{v}}(\mathbf{u}) \) and can be calculated using the formula:
\[ \text{proj}_{\mathbf{v}}(\mathbf{u}) = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \right) \mathbf{v} \]
where \( \cdot \) represents the dot product and \( \|\mathbf{v}\| \) represents the magnitude of \( \mathbf{v} \).
An example of a useful application of vector projection :
In physics, specifically in the field of mechanics. When studying the motion of objects on inclined planes, the vector projection can be used to determine the component of the gravitational force acting on the object that is parallel to the plane. This component, known as the "force of gravity along the incline," is crucial in analyzing the motion of objects sliding or rolling on inclined surfaces. By finding the vector projection of the gravitational force onto the incline, one can determine the force that contributes to the object's acceleration along the incline, allowing for accurate calculations and predictions.
