Spatial vector and three plane projections

Parametric equations

A spatial vector

has length $\Vert \mathbf{v}\Vert =\sqrt{v_x^2+v_y^2+v_z^2}$. Its orthogonal projections onto the coordinate planes are

The three projections are the shadows of $\mathbf{v}$ on the $xy$, $xz$, and $yz$ planes. Repeated components check one another, so the same vector in space can be read from different planes.

Interaction

• Component vx / Component vy / Component vz: drag the three components. Each shadow keeps only two of them
• Three shadows: show all three projections, and check that the repeated components belong to the same $\mathbf{v}$
• xy plane / xz plane / yz plane: switch to one plane. The projection sets the component that is not on that plane to $0$
• Drag the figure to rotate the view and check that each shadow lies on its own coordinate plane. Horizontal view and Elevation do the same rotation

What to notice

• An orthogonal projection onto a coordinate plane sets one component to zero. The $xy$ projection keeps $x,y$, the $xz$ projection keeps $x,z$, and the $yz$ projection keeps $y,z$.
• Each projection is a lower-dimensional reading. Together, the repeated components check one another, so one view does not misread the direction in space.
• $\Vert \mathbf{v}\Vert$ is usually longer than any one projection. The more the vector leans across space, the more the three shadows are needed together.

Related

• Where a line meets a plane
• Plane normal and distance from a point
• Vector projection
• Spatial vectors, planes, and lines

Further reading

• Euclidean vector (Wikipedia)
• Projection (linear algebra) (Wikipedia)