Mathematics 11–12 · Year 12

A thrown ball as a vector-parametrised curve, with skew lines measured in the room

Further work with vectors: Vector equations of lines and curves; Vectors and geometry (Mathematics Extension 2, Year 12)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

A position vector that depends on a parameter traces a curve, while a point and a direction vector describe a straight line, and the scalar product then settles how far a point lies from a line and whether two lines in three dimensions meet or pass each other.

Safety card

Low riskAn adult supervises

Hazards

lines stretched at head height
working above floor level to fix the upper ends
a ball thrown indoors near windows and screens

Controls

flag each line along its whole length with high-visibility tape, keep everyone out of the space the lines cross while they are up, and take them down before the room is used for anything else
use a step platform rather than a chair, with a second person steadying it
clear the throwing line, use a foam or tennis ball, and throw away from windows and screens

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; the work at height and the trip hazard from stretched lines belong in the school's risk assessment before the lesson.

What you need

A tennis ball or a foam ball and a cleared indoor space at least 6 m long
A phone camera on a tripod and a 2.00 m marked pole standing in the frame as the scale
Tracker on a laptop for digitising the flight, and the GeoGebra 3D Calculator for plotting the curve and the two lines
Masking tape and a marker for a target cross on the wall
A tape measure reading to 1 mm over 8 m, and a plumb line
Two 8 m lengths of non-stretch builder's line and four clamps or cup hooks to fix them across the room at different heights
High-visibility tape to flag each stretched line, and a step platform for fixing the upper ends

How to do it

Choose one floor corner of the room as the origin, the wall to its right as the x axis and the wall to its left as the y axis, with z measured upward. Record the room's dimensions in metres.
Film a gentle throw beside the 2.00 m pole, digitise the ball frame by frame in Tracker, and, in the vertical plane of the throw with i horizontal along the throw and j vertically up, fit r(t) = u cos(theta) t i + (h + u sin(theta) t - 4.9 t^2) j to the digitised points.
Differentiate the fitted components to get the velocity vector and evaluate it at launch and at two later frames. Use the scalar product to find the angle between two of those velocity vectors.
Eliminate the parameter to obtain the Cartesian equation of the path, then plot it over the digitised points and state the vertex and the landing point.
Take three digitised points from the flight and test them for collinearity with vectors. Record what the result says about describing the path by a single direction vector.
Write the line of the launch velocity as r = a + lambda b. Measure the position of the target cross, then use the projection of AP onto b to find the perpendicular distance from the cross to that line and the coordinates of the foot of the perpendicular. Check the distance with the tape.
Fix the first builder's line from the origin corner at ceiling height to the far corner near the floor, and the second from a low point on one wall to a higher point on the other. Measure both ends of both lines with the tape and the plumb line to the nearest 10 mm.
Write each line as r = a + lambda b, test whether the direction vectors are parallel, then solve the three component equations. Where they have no common solution and the directions are not parallel, state that the lines are skew.
Find the closest approach by making the joining vector perpendicular to both directions: solve d . b1 = 0 and d . b2 = 0 for the two parameters, compute the two closest points, and take the length of d. Flag those two points on the lines with tape and measure between them.
Repeat with two horizontal lines fixed at different heights, where the shortest distance is the height difference, and confirm the method returns the measured value.

What you should see

A throw digitised at u = 5.0 m/s, theta = 45 degrees and h = 1.20 m gives r(t) = 3.54 t i + (1.20 + 3.54 t - 4.9 t^2) j; eliminating t gives y = 1.20 + x - 0.392 x^2, a parabola with its vertex at (1.276, 1.838) that meets the floor at x = 3.441 m. The velocity vector is (3.54, 3.54 - 9.8 t) m/s, so the speed falls from 5.00 m/s at launch to 3.79 m/s at t = 0.500 s, where the path runs 21.1 degrees below the horizontal, and the scalar product puts 66.1 degrees between those two velocity vectors. Three digitised points from the flight fail the collinearity test, which is the point of the test: no single direction vector describes a curved path. The line of the launch velocity, r = (0, 1.20) + lambda (1, 1), passes 2.263 m from a cross at (2.50, 0.50), with the foot of the perpendicular at (0.900, 2.100), low enough to reach from the step platform in a room whose ceiling is at 2.40 m. Two lines measured from P(0, 0, 2.40) to Q(6.00, 4.00, 0.40) and from R(0, 1.50, 0) to S(4.50, 0, 1.50) give r1 = (0, 0, 2.40) + lambda (3, 2, -1) and r2 = (0, 1.50, 0) + mu (3, -1, 1); the directions are not parallel and the three component equations have no common solution, so the lines are skew. Solving d . b1 = 0 and d . b2 = 0 gives lambda = 0.702 and mu = 0.737, closest points (2.11, 1.40, 1.70) m and (2.21, 0.76, 0.74) m, a joining vector (-0.107, 0.641, 0.961) m and a shortest distance of 1.160 m. Moving the second line's starting point to y = 3.60 m makes the same pair intersect, at (3.60, 2.40, 1.20). Two horizontal lines at 0.60 m and 2.40 m return a shortest distance of 1.800 m. The learner knows it worked when the fitted parabola sits within a ball radius of every digitised point, when the computed perpendicular distance matches the tape measurement between the cross and the foot of the perpendicular within 30 mm, when the tape between the two flagged points on the skew lines agrees with the computed 1.160 m within 30 mm, and when the horizontal-line case reproduces the measured height difference.

What changes

What you change: the parameter along the flight, and the parameters along each stretched line
What you measure: position vector, speed, and the separation of the two lines
What you keep the same: one coordinate frame fixed for the whole lesson
camera position and the 2.00 m scale in every frame
each line kept taut so it stays straight
every coordinate read to the nearest 10 mm

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A vector equation names one point; the parameter runs over all the real numbers, so the one equation names every point on the line.
Two lines that do not meet must be parallel; in three dimensions they can be skew, neither parallel nor intersecting.
The ball travels along the line of its launch velocity; that line is the tangent at launch and the path falls away from it at once.
The distance from a point to a line is the distance to the nearest marked point on it; it is the length of the perpendicular.
Changing a or b changes the line; any point on the line with any non-zero multiple of the direction describes the same line.
Two lines that cross in a photograph intersect; a photograph flattens the depth that decides the question.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Extension 2 11–12 Syllabus (2024), Year 12 focus area Further work with vectors; Extension 2 is a Year 12 course taught from Term 4 2026 with the first HSC examination in 2027, and it replaces the Mathematics Extension 2 Stage 6 Syllabus (2017) taught until then; page read 2026-09-23ME2-12-02
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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