Linear Algebra - How to mentally visualize in 3d space a specific combo of vectors (Gilbert Strang's book - Introduction to Linear Algebra)

Question

1.1 - Vectors & Linear Combinations (Introduction to Linear Algebra (5e) - Gilbert Strang)

(Pg 16) Suppose the vectors u, v, w are in three-dimensional space:

1. What is the picture of all combinations cu?
2. What is the picture of all combinations cu + dv?
3. What is the picture of all combinations cu+ dv + ew?

If they are typical nonzero vectors (components chosen at random), here are the three answers :

1. The combinations cu fill a line through (0, 0, 0).
2. The combinations cu+ dv fill a plane through (0, 0, 0).
3. The combinations cu+ dv + ew fill three-dimensional space.

This is the typical situation! Line, then plane, then space. But other possibilities exist. When w happens to be cu + dv, that third vector w is in the plane of the first two. The combinations of u, v, w will not go outside that uv plane. We do not get the full three-dimensional space.

Query - I'm not getting how to mentally visualize this specific case when w = cu + dv and " The combinations of u, v, w will not go outside that uv plane. "

Answer 1

You asked

how to mentally visualize this specific case when w = cu + dv and "The combinations of u, v, w will not go outside that uv plane."

Consider, e.g., $u=(1,0,0), v=(0,1,0), $ and $w=1u+1v=(1,1,0)$.

Any combination of $u, v, $ and $w$ has $z$-component $0$ and is in the $xy$-plane (which is the $uv$-plane).