Vector Lab
Skill goal: Add, scale, and dot vectors visually: see the parallelogram rule, scalar stretching, and what the dot product measures.
1 · Observe
Blue = u, teal = v, navy = u+v (parallelogram), gold = k·u.
2 · Manipulate
3 · Predict
With u = (3,1), v = (1,4): before computing, guess the sign of u·v. Now drag v to (−4, 1) — what happens to the dot product and the angle between them?
4 · See
5 · Explain
Adding vectors tip-to-tail (or via the parallelogram rule) combines their pushes. The dot product u·v = |u||v|cosθ measures agreement: positive when they point the same way, zero when perpendicular, negative when opposed. A scalar k stretches (k > 1), shrinks (0 < k < 1), or flips (k < 0) a vector.
6 · Challenge
Q: If u·v = 0 (and neither vector is zero), then u and v are…