Predict → set conditions → observe → record → change one variable. Loading an example clears records and creates a reproducible paused state.
01 · Why lengths do not simply add
Vector addition accounts for direction; perpendicular vectors combine by the Pythagorean theorem.
|\vec A+\vec B|=\sqrt{(A_x+B_x)^2+(A_y+B_y)^2}
Try itLoad A=(3,0), B=(0,4), then switch between head-to-tail and parallelogram views.
Prediction. The result is (3,4), magnitude 5 u, not 7 u; direction is about 53.13°.
Compare and explain. When would adding the magnitudes to get 7 u be valid?
Model limit. Distinguish the sum of magnitudes from the magnitude of the vector sum.
02 · Why components add
Components on common axes add as signed numbers and must match the geometric endpoint.
R_x=A_x+B_x,\qquad R_y=A_y+B_y
Try itOpen Components for A=(3,2), B=(−1,3), and inspect each signed component.
Prediction. The result is (2,5), with magnitude √29≈5.385 u.
Compare and explain. Why can the y-component grow while the x-components partly cancel?
Model limit. Components must refer to the same axes; numbers from different coordinate systems cannot be added directly.
03 · Reverse the addition order
For free vectors, changing the addition order changes the path but not the endpoint.
\vec A+\vec B=\vec B+\vec A
Try itRecord (3,2)+(−1,3), then exchange the component pairs of A and B and record again.
Prediction. Both result component pairs are (2,5).
Compare and explain. Are the path lengths and every intermediate position also identical?
Model limit. Equal net displacement does not imply identical motion histories.
04 · Subtraction as adding the opposite
Subtracting B means adding an equal-magnitude vector with the opposite direction.
\vec A-\vec B=\vec A+(-\vec B)
Try itFor A=(3,2), B=(−1,3), switch between addition and subtraction and record.
Prediction. The sum is (2,5); the difference is (4,−1), magnitude √17≈4.123 u.
Compare and explain. How do B−A and A−B compare in magnitude and direction?
Model limit. Dot and cross readouts use original A and B, distinct from the negated construction arrow.
05 · Cancellation and the zero vector
Equal and opposite vectors sum to zero. Its magnitude is zero, but its direction is not 0°; it is undefined.
\vec B=-\vec A\Rightarrow\vec R=\vec 0
Try itLoad Cancel and inspect result direction. Slightly change B’s y-component from −2 to −1.9.
Prediction. Exact cancellation shows — for direction; after the change the result is (0,0.1), at 90°.
Compare and explain. Can a tiny perturbation strongly change direction while magnitude stays small?
Model limit. Distinguish rounded displays from exact zero; zero vectors receive no direction arc.
06 · Scale one vector
Positive scaling keeps direction and changes magnitude; the resultant endpoint moves on a line through A’s tip parallel to B.
\vec R=\vec A+k\vec B
Try itFor A=(3,0), B=(−1,2), use advanced settings to record k=0, 1, 2 and 3.
Prediction. The endpoints are (3,0), (2,2), (1,4) and (0,6).
Compare and explain. Why is resultant magnitude not proportional to k?
Model limit. k is dimensionless; a dimensional multiplier may change the kind of physical quantity.
07 · Dot product reads alignment
The dot product measures alignment through sign and magnitude; it vanishes for perpendicular vectors.
\vec A\cdot\vec B=A_xB_x+A_yB_y=|\vec A||\vec B|\cos\theta
Try itKeep A=(3,0) and record B=(2,0), (0,2) and (−2,0).
Prediction. The dot products are 6, 0 and −6 u².
Compare and explain. For force and displacement, how would the dot-product sign relate to work?
Model limit. Both vectors here use u. Actual work requires force and displacement units separately.
08 · Signed cross product and area
Cross-product magnitude is parallelogram area, while its sign distinguishes order.
(\vec A\times\vec B)_z=A_xB_y-A_yB_x
Try itRecord A=(3,0), B=(0,4) in Parallelogram view, then exchange A and B.
Prediction. Both areas are 12 u²; the signed z-component changes from +12 to −12.
Compare and explain. Why is addition unchanged by swapping order while the cross product reverses?
Model limit. This is the out-of-plane z-component; geometric area is its nonnegative absolute value.
09 · Translate without changing vectors
Free vectors are unchanged by translating the tail while keeping components fixed; this permits geometric construction.
\Delta\vec r=(x_2-x_1,\ y_2-y_1)
Try itDrag the common origin or change its coordinates in advanced settings; compare result, dot and cross values.
Prediction. All readouts remain unchanged when A and B components are unchanged.
Compare and explain. Would translating the application point of a force preserve its rotational effect?
Model limit. Torque due to force application points is outside this free-vector model.