Eleventh Grade · Science · week 2

Scalars and Vectors: Graphical Addition

Distinguish scalar from vector quantities and add vectors graphically using the Pythagorean theorem for perpendicular cases.

Some quantities in physics are fully described by a single number: mass, time, distance, and speed only need a size (a "how much"). These are scalars. Other quantities need a size AND a direction to mean anything: displacement, velocity, and force are vectors. Saying a plane flies "500 km/h" is a speed (scalar); saying it flies "500 km/h due north" is a velocity (vector). Mixing these up is one of the most common errors in physics, so we will be careful all year to ask "is a direction part of this answer?"

A vector is drawn as an arrow: its length represents magnitude (to some scale) and the way it points represents direction. To add two vectors graphically, use the head-to-tail method -- draw the first vector, then start the second vector's tail where the first vector's head ended. The resultant (the sum) is the single arrow from the very start to the very end. When two vectors are exactly perpendicular, the resultant's magnitude comes from the Pythagorean theorem, since the two vectors and the resultant form a right triangle.

📋 For the teacher
Teach it (10–15 min):
  • Walk out real displacements together -- pace off 3 steps one way and 4 steps a perpendicular way, then measure the straight-line distance back to the start with a tape measure. It will land close to 5 steps, and seeing it beats being told it.
  • Keep a running list on the fridge of scalars versus vectors as new quantities come up all year (mass, force, energy, momentum...) -- sorting each new quantity into the right column builds the habit for good.
  • For head-to-tail addition, use two pencils or two strips of paper as physical arrows they can pick up, rotate, and place tail-to-head on the table before drawing anything.
  • When the two vectors are NOT perpendicular, just note for now that the Pythagorean theorem will not directly apply -- next week's component method handles that case cleanly.
Talk about it:
  • What is the key difference between how you describe a scalar and how you describe a vector?
  • Why does adding two perpendicular vectors turn into a Pythagorean-theorem problem?
  • Can you think of a vector and a scalar that are easy to confuse in everyday speech?

Mastery looks like: They correctly classify quantities as scalar or vector and find the magnitude of the resultant of two perpendicular vectors using the Pythagorean theorem.

Common stumbles: Adding vector magnitudes directly even when the vectors point in different directions; forgetting that a full vector answer needs a direction, not just a magnitude.

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