Orthogonal Projection of a Vector Onto a Line

Linear Algebra

Projections

Medium

Project

(4, 2)

orthogonally onto the line spanned by

(1, 1)

Hint 1

The projection formula is $proj_{v} (x) = \frac{x \cdot v}{v \cdot v} v$ .

Hint 2

Compute $x \cdot v$ and $v \cdot v$ for $x = (4, 2)$ and $v = (1, 1)$ .

Show worked solution

1. Compute the dot products
With $x = (4, 2)$ and $v = (1, 1)$ , $x \cdot v = 6$ and $v \cdot v = 2$ .
2. Apply the projection formula
Scaling $v$ by $x \cdot v / v \cdot v$ gives the projection.
$proj_{v} (x) = \frac{6}{2} (1, 1) = (3, 3)$

Answer: (3,3)

1. Compute the dot products