Precalculus Lesson

Vectors: From Head-to-Tail to Dot Product Work Projections

Shared by A STEM Educator

11th GradeBTC Style / Thinking Classrooms50 min

Learning Objective

Students will represent vectors in component form, perform linear combinations, scale unit vectors, and compute dot product work projections — connecting geometric intuition to algebraic structure throughout.

Lesson Overview

Students progress from geometric head-to-tail vector addition through algebraic component form, linear combinations, unit vector scaling, and culminate with dot product work projections in force/navigation contexts. Each task changes exactly one layer of complexity.

Materials

Vertical whiteboards or large paper
Dry-erase markers (multiple colors)
Pencil and paper
Phone or basic calculator

Scaffolded Task Progression

1Task 1

Two forces act on a ship:

F_{1}

points due East with magnitude 3 units, and

F_{2}

points due North with magnitude 4 units. On your whiteboard, draw both vectors head-to-tail (place the tail of

F_{2}

at the head of

F_{1}

). Draw the resultant

R = F_{1} + F_{2}

. What is the magnitude of

R

? What angle does

R

make with due East?

2Task 2

The same ship problem, but now express each force in component form before adding. Write

F_{1} = ⟨ 3, 0 ⟩

and

F_{2} = ⟨ 0, 4 ⟩

. Add them algebraically: what is

R = F_{1} + F_{2}

in component form? Verify the magnitude matches Task 1.

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