Finding Orthogonal Vector Component of u with respect to v

Find $w_2$, the vector component of $\mathbf{u}$ orthogonal to $\mathbf{v}$, where $\mathbf{u} = (3, 5)$ and $\mathbf{v} = (2, 4)$.

In this problem, we are asked to find the vector component of one vector that is orthogonal to another. This concept is central to vector mathematics, which is the study of quantities that have both magnitude and direction. When dealing with vectors in two-dimensional space, we frequently explore relationships between multiple vectors, including projections and orthogonal components. Understanding these relationships can help us manipulate and interpret vector equations more effectively.

To intuitively approach this problem, we first recognize the significance of orthogonality. Two vectors are orthogonal if they are perpendicular to each other, resulting in a dot product of zero. Our objective is to ascertain which part of vector u neither aligns with nor lies in the same direction as vector v, but rather is perpendicular to it. This orthogonal vector component is essential in applications like decomposing forces in physics problems or determining directions of least resistance.

Solving for the orthogonal component involves a strategic use of vector projection. We typically find the component of u along v and subtract it from u to obtain the orthogonal part. This operation leverages the properties of the dot product and vector addition. Hence, learning to determine such components enhances our problem-solving toolkit, especially for more advanced vector calculus.

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