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A unit vector is a vector with a magnitude of one. It is generally used to specify a direction. The simplicity of unit vectors enables them to exclusively represent direction, making them extremely useful in problems that focus on vector directionality. Any vector can be turned into a unit vector by dividing it by its own magnitude.
Key features of unit vectors include:

• Their length is always 1, which simplifies calculations involving them.
• They maintain the direction of the original vector.
• Unit vectors are often denoted with a "hat", for example, \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) in three dimensions, which point along the axes of a Cartesian coordinate system.

In our problem, \( \vec{a} \) is a unit vector, meaning that when we dot it with itself, we obtain 1, as \( \vec{a} \cdot \vec{a} = 1 \). Using a unit vector simplifies the calculation and interpretation of vector projections, as seen in calculating \( \vec{c} \).

Vector components are the projections of a vector along the axes of a coordinate system. They enable the simplification of vector arithmetic and geometry by reducing the vector into manageable parts. By expressing a vector in terms of its components, we can perform operations on each part separately.
Understanding how vector components work:

• A vector can be broken down into its horizontal and vertical components in 2D or further split along the \( x, y, \) and \( z \) axes in 3D.
• Components are essential for projecting vectors onto other vectors, which simplifies the understanding of vector interactions.

In the exercise, \( \vec{b} \) has a component \( \vec{c} \) along the direction of \( \vec{a} \). This is calculated as \( \vec{c} = (\vec{b} \cdot \vec{a}) \vec{a} \). The component \( \vec{d} \), on the other hand, is perpendicular to \( \vec{a} \). Together, these components fully represent \( \vec{b} \) in relation to \( \vec{a} \), showing both its direction along and away from \( \vec{a} \).

Perpendicular vectors are vectors that meet at a right angle. This means their dot product equals zero. Understanding this property of perpendicular vectors is crucial as it can simplify the problem-solving process when analyzing vector relationships.
Key concepts about perpendicular vectors:

• If two vectors \( \vec{u} \) and \( \vec{v} \) are perpendicular, their dot product \( \vec{u} \cdot \vec{v} = 0 \).
• Perpendicular vectors are orthogonal, meaning they hold no component in each other's direction.

In the problem at hand, proving \( \vec{d} \cdot \vec{a} = 0 \) demonstrates that \( \vec{d} \) is perpendicular to \( \vec{a} \). This is achieved by showing \( \vec{d} \) contains only the part of \( \vec{b} \) that does not point in the direction of \( \vec{a} \). This perpendicularity confirms that \( \vec{d} \) lies entirely in the plane orthogonal to \( \vec{a} \).