C. Geometry and Algebra of Vectors · Vectors in Two-Space and Three-SpaceVectors: Quantities with Magnitude and Direction
C1.1About 65 minutes
Learning goals- I can explain the difference between a scalar and a vector and give examples of each.
- I can describe a vector by its magnitude and direction and draw it as a directed line segment.
- I can tell distance from displacement and speed from velocity.
- I can describe real-world uses of vectors such as displacement, forces in structures, computer animation and GPS velocity.
Words to know
scalar
A quantity with magnitude only, such as mass, time, temperature, distance or speed.
vector
A quantity with both magnitude and direction, such as displacement, velocity, force or acceleration.
magnitude
The size of a vector, written |𝐮|. It is always zero or positive.
directed line segment
An arrow from a tail point to a head point, used to represent a vector.
displacement
The vector from a starting position to a final position.
position vector
The vector from a fixed origin to a point; it locates the point exactly.
Where exactly are you?
Your friend texts: "I'm 69 km from Lindsay, Ontario." Can you find them? Not yet. They could be north of Lindsay, south of it, or anywhere else on a circle of radius 69 km around the town. To pin down a location you need two pieces of information: how far and which way.
That pair of ideas, a size and a direction, is the heart of this unit. A quantity that needs both is called a vector. Vectors describe motion, forces, wind, the position of a drone, the way a video-game character slides across a screen and much more. In the next three units of this course (this one, Dot and Cross Products and Lines and Planes) vectors become the main language for geometry in two and three dimensions.
Scalars and vectors
A scalar is a quantity that is completely described by a single number (with a unit). Examples: mass (60 kg), time (3.5 h), temperature (−12 °C), distance (7 km), speed (90 km/h), energy (250 J).
A vector is a quantity that has both magnitude (size) and direction. Examples:
- displacement: 5 km [N 36.9° E] (how far and which way you ended up from where you started)
- velocity: 90 km/h [W] (speed with a direction)
- force: 600 N straight down (your weight is a force)
- acceleration: 9.8 m/s² downward
Speed and velocity are a famous pair: speed is a scalar, velocity is a vector. A car going around a roundabout at a steady 30 km/h has constant speed but its velocity keeps changing, because its direction changes.
Remember from Grade 11: if you took Physics (SPH3U), the Kinematics: Motion in One and Two Dimensions unit used scalars and vectors for motion, wrote directions like [N 30° E] and added vectors tip to tail. If you didn't, everything you need is built from scratch here.
How we write vectors
A vector is drawn as an arrow, also called a directed line segment. The length of the arrow shows the magnitude (using a scale, such as 1 cm : 10 N), and the arrow points in the vector's direction. The starting point is the tail and the arrow point is the head (or tip).
On this site, vectors are printed in bold: 𝐮, 𝐯, 𝐅. The vector from point A to point B is 𝐀𝐁. In your notebook, draw a small arrow over the letters instead, since bold is hard to do by hand. The magnitude of 𝐮 is written |𝐮| and is always a non-negative scalar. For example, if 𝐅 is a force of 600 N downward, then |𝐅| = 600 N.
Is it a vector? Ask two questions
- Does the quantity have a size? (Every scalar and vector does.)
- Does it make sense, and matter, to ask "in which direction?"
If the answer to question 2 is yes, it is a vector. "Which direction is 40 kg?" makes no sense, so mass is a scalar. "Which direction is the wind blowing?" matters a lot, so wind velocity is a vector.
Worked exampleExample 1: Sorting quantities
Classify each as a scalar or a vector: (a) 12 kg of flour; (b) a 300 N pull on a sled rope at 30° above the ground; (c) 25 km/h; (d) a wind of 25 km/h from the northwest; (e) 40 J of energy; (f) −15 °C.
- Look for a direction stated or implied.
- (b) has a direction (30° above the ground) and (d) has a direction (from the northwest). These are vectors.
- (a), (c), (e) and (f) are sizes only. A negative sign on a temperature is not a direction: −15 °C is just a number below zero on a scale.
Vectors: (b) and (d). Scalars: (a), (c), (e), (f).
Worked exampleExample 2: The Lindsay problem
Explain why knowing that someone is 69 km from Lindsay, Ontario, is not enough to identify their exact position. What extra information would fix it?
- "69 km from Lindsay" gives a distance only, which is a scalar.
- Every point on a circle of radius 69 km centred on Lindsay is 69 km away, so there are infinitely many possible positions.
- A position is described by a position vector: a magnitude and a direction from a fixed reference point (an origin).
Add a direction, for example "69 km from Lindsay at N 40° E". The magnitude and direction together form a vector that picks out exactly one point.
Worked exampleExample 3: Distance versus displacement
Maya snowshoes 3 km east and then 4 km north. Find the distance she travels and her displacement.
- Distance just adds the path lengths: 3 + 4 = 7 km.
- The displacement is the arrow from start to finish. The two legs are perpendicular, so by the Pythagorean theorem the magnitude is √(3² + 4²) = √25 = 5 km.
- Direction: measure the angle from north toward east. In the right triangle, the side opposite this angle is 3 (east) and the adjacent side is 4 (north), so tan θ = 3/4 and θ = tan⁻¹(0.75) ≈ 36.9°.
Distance 7 km (scalar). Displacement 5 km [N 36.9° E] (vector).
Maya's path (solid arrows) is 7 km long, but her displacement (dashed arrow) is 5 km at 36.9° east of north.Vectors in the real world
The curriculum asks you to notice where vectors are used. A few examples:
- Structural design. Engineers treat every force on a bridge or roof truss as a vector. A beam is safe only if the forces on each joint add up to zero (you will see this in the lesson on forces).
- Computer graphics and animation. To slide a character across the screen, a program adds the same translation vector, such as [4, 2] (4 pixels right, 2 up), to every point of the drawing in each frame.
- GPS and navigation. A GPS receiver reports your speed and your heading. Together they give your velocity vector. Pilots combine the plane's velocity with the wind's velocity to find where the plane will really go.
- Weather. Environment and Climate Change Canada reports wind as a speed and a direction, such as "wind northwest 30 km/h". By convention, the direction is where the wind blows from.
Just for funWhy did the vector break up with the scalar? The scalar never knew where the relationship was going.
Hands-on activityTreasure hunt: distance versus direction
You need: a small object to hide (a spoon or a sock), a metre stick or tape measure, a phone compass app (or a paper arrow marked N that a grown-up lines up with north for you), and a partner.
- Stand on a starting spot (mark it with a sock or a piece of tape). Your partner closes their eyes while you hide the object somewhere within about 5 m of the spot.
- Give your partner only a distance: "It is 3.2 m from the start." Ask them to name all the places it could be. (Anywhere on a circle!)
- Now give a vector: "It is 3.2 m from the start, in the direction N 60° E" (use the compass to find the direction). Your partner should walk straight to it.
- Switch roles. Then try giving two vectors in a row ("2 m north, then 1.5 m west") and have your partner find the single vector from the start to the object.
What to look for: the distance-only clue gives a whole circle of possibilities; the vector gives exactly one spot. For two vectors in a row, the single "start to finish" vector should be shorter than 2 + 1.5 = 3.5 m (about 2.5 m), because the legs are at right angles.
Your turn
1Classify as scalar or vector: (a) 4.5 L of water; (b) a car's velocity of 100 km/h [E]; (c) a 9.8 N force pulling straight down; (d) 21 °C; (e) 18 km of trail.
Vectors: (b), (c). Scalars: (a), (d), (e).
2A hockey puck slides 20 m from the goal line. Is "20 m" enough to describe where it ended up? Explain.
No. 20 m is a distance (scalar). The puck could be anywhere on a circle of radius 20 m around its starting point (that is still on the ice). You need a direction too, which makes it a displacement vector.
3Explain the difference between speed and velocity using a car on a circular racetrack moving at a constant 120 km/h.
Speed is the scalar 120 km/h and stays the same. Velocity is a vector; its direction keeps changing as the car turns, so the velocity changes even though the speed doesn't.
4A cyclist rides 5 km west, then 12 km south. Find the distance travelled and the magnitude of the displacement.
Distance = 5 + 12 = 17 km. Displacement magnitude = √(5² + 12²) = √169 = 13 km.
5For the cyclist above, find the direction of the displacement as an angle measured from south toward west.
tan θ = 5/12, so θ = tan⁻¹(5/12) ≈ 22.6°. The displacement is 13 km [S 22.6° W].
6A vector 𝐅 represents a force of 250 N pushing east. What is |𝐅|? What is |𝐅| if the push is toward the west instead?
|𝐅| = 250 N in both cases. Magnitude is never negative and does not depend on direction.
7Give one example of how vectors are used in computer animation.
For example: moving an image across the screen by adding a translation vector such as [4, 2] to the coordinates of every point in each frame. (Other good answers: velocity of a game character, direction a camera is pointing.)
More practice: warm-up
1Is time a scalar or a vector?
Scalar. It has no direction.
2Is weight (the force of gravity on you) a scalar or a vector?
Vector. It points down, toward the centre of Earth.
3What are the two parts of every vector?
Magnitude (size) and direction.
4Can the magnitude of a vector be negative?
No. A magnitude is a length, so it is always zero or positive.
5In an arrow diagram, what does the length of the arrow show?
The magnitude of the vector (using the diagram's scale).
More practice: on your own
1A drone flies 600 m north, then 800 m east. Find its displacement (magnitude and direction from north toward east).
Magnitude √(600² + 800²) = √1 000 000 = 1000 m. Angle from north: tan θ = 800/600, θ ≈ 53.1°. Displacement 1000 m [N 53.1° E].
2A runner does 4 laps of a 400 m track and stops exactly where she started. Find her distance and her displacement.
Distance = 4 × 400 = 1600 m. Displacement = 0 m, because she ends where she started (the zero vector has no direction).
3Environment Canada reports "wind west 40 km/h". In which direction is the air moving?
Wind directions say where the wind comes from, so the air moves toward the east at 40 km/h.
4On a scale of 1 cm : 50 N, how long is the arrow for a force of 175 N?
5Two forces both have magnitude 80 N. Must they be equal vectors? Explain.
No. Equal vectors need the same magnitude and the same direction. If one pulls east and the other pulls north, they are different vectors with the same magnitude.
Challenge
1A boat leaves a dock and travels 9 km east, then 4 km north, then 3 km west. Find its displacement from the dock (magnitude and direction from east toward north).
Net east: 9 − 3 = 6 km. Net north: 4 km. Magnitude √(6² + 4²) = √52 = 2√13 ≈ 7.2 km. Angle from east toward north: tan θ = 4/6, θ ≈ 33.7°. So about 7.2 km [E 33.7° N].
2A point P is 10 km from a town T. List every possible distance between P and a second point Q that is 4 km from T. Explain using vectors or a sketch.
Q can be anywhere on a circle of radius 4 km around T. The distance PQ is smallest when Q lies on segment TP (10 − 4 = 6 km) and largest when Q is on the opposite side (10 + 4 = 14 km). Every distance from 6 km to 14 km is possible. Distances alone leave a whole range; directions would fix the answer.
3Explain why |𝐮| = |𝐯| does not mean 𝐮 = 𝐯, but 𝐮 = 𝐯 does mean |𝐮| = |𝐯|.
If 𝐮 = 𝐯, they have the same magnitude and the same direction, so in particular |𝐮| = |𝐯|. But two vectors can have the same magnitude and point in different directions (for example 5 km [N] and 5 km [E]), so equal magnitudes do not force equal vectors.
Quiz
1. Which of these is a vector?
2. Someone is 69 km from Lindsay. Why can't you find their exact location?
3. A runner jogs 6 km north and then 8 km east. What is the magnitude of the displacement?
4. Which statement about speed and velocity is true?
5. How is the vector from point A to point B written on this site?
6. The wind is reported as "north 20 km/h". Which way is the air moving?
Big idea (C1.1): a vector has a magnitude and a direction. Distance, speed, mass and time are scalars; displacement, velocity, force and acceleration are vectors.
How to help: Ask everyday questions and let your teen sort the answers: "The thermostat says 21 degrees: scalar or vector?" "Our car is going 50 km/h west: which part is the direction?" The Lindsay question in Example 2 is a good one to discuss at the dinner table. The activity needs only a measuring tape and a compass app.
Common difficulties:
- Thinking a negative sign makes something a vector (−15 °C is still a scalar).
- Using "speed" and "velocity" as if they were the same word.
- Writing |𝐮| as negative. Magnitudes are never negative.
- Forgetting that wind is reported by the direction it blows from.
Notation: the site prints vectors in bold (𝐮). By hand, students write an arrow over the letter, as their teachers and textbooks do.