Question

A vector u and a line Lin R² are given. Compute the orthogonal projection w of u on L, and use it to compute the distance d from the endpoint of u to L.

[5]
u= [ ] and y = 0
[0]

Answer 1

To find the orthogonal projection of a vector onto a line, one must consider the x-component of the vector. For a line described by y = 0, the projection is along the x-axis. The distance from the endpoint of the vector to the line is the absolute value of its y-component.

Step-by-step explanation:

To compute the orthogonal projection of a vector u onto a line L in R², which is represented by the equation y = 0, we first need to understand that the projection w on the x-axis is simply the x-component of vector u.

Since the line L is the x-axis itself, the projection of u is pointing in the direction of the x-axis with the same magnitude as the x-component of u.

To find the distance d from the endpoint of u to the line L, we can use the y-component of u, since it represents the shortest distance to the line which is parallel to the y-axis.

For example, if vector u has components (3, 4), the orthogonal projection w onto the x-axis (line L) would be (3, 0). The distance d would then be the absolute value of the y-component of u, which is 4.