Projection of a Vector onto Another Vector
To find the projection of (v) onto (u), calculate the component of (v) that points in the same direction as (u). In Euclidean space, the vector projection is
[ \boxed{\operatorname{proj}_{u}v=\frac{v\cdot u}{u\cdot u}u} ]
where (u\neq 0). Now, the result is the vector on the line spanned by (u) that is closest to (v). Understanding this formula requires familiarity with the dot product, vector magnitude, and the geometric idea of perpendicular distance.
What Is the Projection of (v) onto (u)?
The projection of (v) onto (u) is the part of (v) that lies along the direction of (u). Any vector (v) can be divided into two components relative to a nonzero vector (u):
[ v=\operatorname{proj}{u}v+\operatorname{proj}{u^\perp}v ]
The first component is parallel to (u), while the second is perpendicular to (u). Here, (u^\perp) represents the entire subspace of vectors orthogonal to (u) The details matter here..
The projection is also the closest point to the endpoint of (v) on the line passing through the origin in the direction of (u). Because the difference between (v) and its projection is perpendicular to the line,
[ \left(v-\operatorname{proj}_{u}v\right)\cdot u=0. ]
This perpendicular relationship is the central property used to derive the projection formula.
The Projection Formula
For vectors (u) and (v) in (\mathbb{R}^n), the projection is
[ \operatorname{proj}_{u}v
\frac{v\cdot u}{|u|^2}u. ]
Since
[ |u|^2=u\cdot u, ]
the equivalent form is
[ \boxed{\operatorname{proj}_{u}v
\frac{v\cdot u}{u\cdot u}u}. ]
The formula has three important parts. First, (v\cdot u) measures how strongly (v) points in the direction of (u). Day to day, second, dividing by (u\cdot u) adjusts the result for the length of (u). Third, multiplying by (u) produces a vector parallel to (u) Surprisingly effective..
The requirement (u\neq 0) is essential. If (u) is the zero vector, it does not define a direction, so its projection is undefined.
How to Find the Projection Step by Step
To find the projection of (v) onto (u), follow this procedure. First, compute the dot product (v\cdot u). For vectors (v=(v_1,v_2,\ldots,v_n)) and (u=(u_1,u_2
