Plane vectors

Idea

As a position, the tip of the arrow is a point in the plane.

As a direction, the arrow is something you measure along. A projection and a dot product both ask how much of another vector lies that way.

As coordinates, the same position is a pair of coefficients of basis vectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$. A normal $\mathbf{n}$ and a constant $c$ draw the line $\mathbf{n}\cdot \mathbf{x}=c$. These are plane vectors written as pairs, not columns.

Interaction

• Guide: one arrow $\mathbf{p}$, switched among Position, Direction, and Coordinates
• Direction / projection: drag the tips and watch the projection, the angle, and the sign of the dot product. Unit circle leaves only $\cos \theta$
• Coordinates / basis: write a vector as coefficients of a basis. Coefficient s and Coefficient t change that reading
• Normal / line: a normal turns a direction into the line $\mathbf{n}\cdot \mathbf{x}=c$. Constant c slides the line, which stays perpendicular to $\mathbf{n}$

Suggested order: use Guide for the three readings of one arrow, then the three focused modes for the projection, the basis, and the normal.

What to notice

• The same arrow can be a position, then a direction to measure along, then coordinates in a basis.
• A projection and a dot product measure the component along a chosen direction. Perpendicular means that component is zero.
• Changing the basis changes the coordinates. A normal turns a direction into a line.

Related

• Vector addition and scalar multiplication
• Geometric significance of the dot product
• Vector projection
• Basic patterns of a vector field

Further reading

• Euclidean vector (Wikipedia)
• Dot product (Wikipedia)
• Vector space (Wikipedia)