To find the orthogonal projection of a vector onto another vector (or onto a line spanned by that vector), you use a specific formula involving the dot product and the magnitude squared of the vector you are projecting onto.
The orthogonal projection of a vector v onto a non-zero vector u, often denoted as proj_u v, is given by the formula:
$$ \text{proj}_{\mathbf{u}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{u}|^2} \mathbf{u} $$
This formula determines the component of v that lies in the direction of u.
Understanding the Formula
Let's break down the components of the orthogonal projection formula:
- v: The vector you want to project.
- u: The vector onto which you are projecting v (this vector defines the direction or the line).
- v ⋅ u: The dot product of vectors v and u. This is a scalar value. It measures how much v points in the direction of u.
- ||u||²: The squared magnitude (or squared length) of vector u. This is also a scalar value, calculated as u ⋅ u.
- $$ \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{u}|^2} $$: This entire fraction is a scalar. It represents the scalar projection of v onto u. It tells you how "long" the projection is along the direction of u (signed, meaning it could be negative if v points opposite to u).
- $$ \left( \frac{\mathbf{v} \cdot \mathbf{u}}{|\mathbf{u}|^2} \right) \mathbf{u} $$: The scalar projection is multiplied by the vector u. This scales the direction vector u by the calculated scalar length, resulting in a vector that is parallel to u and has the correct length and direction of the projection.
Steps to Calculate Orthogonal Projection
To calculate the orthogonal projection of v onto u:
- Calculate the dot product v ⋅ u.
- Calculate the magnitude squared of u, which is ||u||² = u ⋅ u.
- Divide the dot product (v ⋅ u) by the magnitude squared (||u||²). This gives you the scalar multiplier.
- Multiply the result from step 3 by the vector u.
The resulting vector is the orthogonal projection of v onto u.
Context from the Reference
The provided reference mentions finding the orthogonal projection of vector x onto a line l where vector u is an orthogonal basis for the line. This aligns perfectly with the formula described above. If a line l is defined by a single vector u, then u serves as an orthogonal basis for that line. Therefore, the projection of vector x onto the line l is simply the projection of x onto the vector u, which is calculated using the formula:
$$ \text{proj}_{\mathbf{u}} \mathbf{x} = \frac{\mathbf{x} \cdot \mathbf{u}}{|\mathbf{u}|^2} \mathbf{u} $$
As stated in the reference, "Here is that vector u because we have a single vector is an orthogonal basis for the line l and therefore the projection formula does apply applying the formula the orthogonal projection of vector x".
Example
Let's find the orthogonal projection of vector v = <4, 2> onto vector u = <3, 0>.
-
Calculate the dot product v ⋅ u:
v ⋅ u = (4 3) + (2 0) = 12 + 0 = 12 -
Calculate the magnitude squared of u:
||u||² = u ⋅ u = (3 3) + (0 0) = 9 + 0 = 9 -
Divide the dot product by the magnitude squared:
Scalar multiplier = (12) / (9) = 12/9 = 4/3 -
Multiply the result by vector u:
proj_u v = (4/3) <3, 0> = <(4/3)3, (4/3)*0> = <4, 0>
So, the orthogonal projection of <4, 2> onto <3, 0> is <4, 0>.
Orthogonal Projection
