Projection and Orthogonal Components of a Vector

For vectors $\mathbf{u} = 6\mathbf{i} - 3\mathbf{j} + 9\mathbf{k}$ and $\mathbf{v} = 4\mathbf{i} - \mathbf{j} + 8\mathbf{k}$, find the two components $w_1$ and $w_2$ of vector $\mathbf{u}$, where $w_1$ is the projection of $\mathbf{u}$ onto $\mathbf{v}$ and $w_2$ is the component orthogonal to $\mathbf{v}$.

In this problem, we delve into the concepts of vector projection and orthogonality within the framework of three-dimensional vectors. When dealing with vector projections, one must understand how a vector can be decomposed into two distinct components: one that is parallel to another vector, and another component that is orthogonal (or perpendicular) to that vector.

The projection, denoted here as $w_1$, is calculated using the dot product, which measures how much of one vector goes in the direction of another, combined with the scalar division by the magnitude squared of the vector being projected onto. This projection is fundamentally about finding the scalar component that aligns one vector with another in space.

The second component, $w_2$, is the orthogonal component, representing the part of the original vector that is perpendicular to the vector onto which it was projected. Understanding this concept requires a grasp of orthogonality and how vector subtraction can be used to isolate this component.

By subtracting the projection from the original vector, you isolate the vector component that does not contribute to the projection, effectively showing how the original vector is split in its multidimensional environment. This problem not only reinforces the understanding of vector operations such as dot products and vector subtraction, but also highlights the geometric interpretations which are critical in multidimensional space analysis, especially when applied to physical phenomena or advanced mathematical problems.

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