Finding Vector Components along a Direction in R3

Given a vector $\mathbf{u}$ and another nonzero vector $\mathbf{a}$ in $\mathbb{R}^3$, find the vector component of $\mathbf{u}$ along $\mathbf{a}$ and the vector component of $\mathbf{u}$ orthogonal to $\mathbf{a}$.

In this problem, you are asked to find two components of a vector in relation to another vector: one component that lies along the given direction and another that is orthogonal to it. The resolution of such a problem typically involves understanding the concept of projections in vector spaces. When working with vectors in three-dimensional space, the notion of decomposing a vector into components that align with and are perpendicular to another vector is crucial. This concept is not only common in linear algebra but also has significant applications in physics, particularly in fields involving forces and motion.

To solve this, you will use the dot product, which provides a mechanism to measure angles and projections in vector spaces. The dot product of two vectors is pivotal when it comes to determining how much one vector goes in the direction of another. By using the formula for projection, which involves the dot product, you can find the vector component of the given vector along another vector. The component of the vector orthogonal to the given vector can be found by subtracting the projection from the original vector.

Understanding these decomposition principles helps improve comprehension of how vectors interact in various directions and enhances problem-solving skills in both theoretical and practical realms. Mastery of this topic lays the groundwork for more complex analyses involving vector spaces, transformations, and subspaces.