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Lesson 1 of 6Vectors
Unit 5 · Lesson 1 of 6

Vectors, Scalars and Direction

A vector carries a size AND a direction. Learn the notation, when two vectors are equal or opposite, and the three ways to state a direction: a standard angle, a true bearing and a quadrant bearing.

5 min read9 practice questionsAsk AI about this →

Key things to remember

  1. 1A scalar has magnitude only; a vector has magnitude and direction. Magnitude .
  2. 2Equal vectors have the same magnitude and direction, regardless of position. .
  3. 3Standard angle: counterclockwise from the positive -axis. True bearing: clockwise from north, three digits.
  4. 4Convert between them with (add if negative).
  5. 5Quadrant bearing: start at N or S, turn toward E or W, angle from to (for example, a true bearing of is N W).

Suppose a friend texts you: "I'm 69 km from Lindsay, Ontario." Where are they? You can't tell. They could be anywhere on a circle of radius 69 km around Lindsay: north toward cottage country, south toward Lake Ontario, or east past Peterborough. The number 69 tells you HOW FAR but not WHICH WAY. Add a direction ("69 km north-west of Lindsay") and the position is pinned down. A quantity that needs both pieces of information is a vector, and this unit is about doing mathematics with them.

Scalars and vectors

A SCALAR is a quantity described completely by a single number (with a unit): mass ( kg), time ( s), temperature ( °C), speed ( km/h), distance, energy, volume. A VECTOR has a magnitude (its size, never negative) and a direction: displacement ( km north-west), velocity ( km/h due east), acceleration, force, weight. A negative sign on a scalar such as °C does not make it a vector. It is a value below zero on a scale, not a direction in space.

Vector: 5 units, three different directions5 [E]5 [N]5 [N 45° E]Scalar5no directionto giveSame magnitude, three genuinely different vectors — the direction is part of the quantity.
Fig 1Three arrows of the same length (5 units) pointing east, north and north-east are three different vectors. A scalar such as 5 has a size and nothing else.
  • Displacement and position: a GPS receiver reports where you are relative to a reference point, which needs a distance and a direction.
  • Velocity: navigation apps combine your speed with your heading, and a pilot must combine the plane's velocity with the wind's.
  • Forces in structural design: each member of a bridge truss pushes or pulls in a particular direction, and an engineer adds those force vectors to check that every joint balances.
  • Computer graphics and animation: a character on screen moves each frame by a small displacement vector; changing the vector changes both how fast and which way it moves.
VideoOntario teacher Mr. Cioffi introduces vectors for MCV4U: what separates a vector from a scalar and how vectors are drawn and named.Watch on YouTube

Notation

A vector is drawn as a directed line segment: an arrow whose length shows the magnitude and whose arrowhead shows the direction. The arrow from point (the TAIL) to point (the HEAD) is written . A vector that is not tied to named points gets a letter with an arrow, (printed sources often use bold v instead). The magnitude is written with absolute-value bars: or , and it is a scalar that is zero or positive. The ZERO VECTOR has magnitude and no defined direction.

Equal vectors and opposite vectors
Two vectors are EQUAL when they have the same magnitude and the same direction, wherever they are drawn. Sliding an arrow around the page without turning or stretching it gives the same vector. The OPPOSITE of , written , has the same magnitude and the reverse direction. Reversing the letters reverses the arrow: .

Three ways to state a direction

Angle from +x axis+x+y30°v30°start on the +x axis,turn counterclockwiseTrue bearingENWS060°v060°start at north,turn clockwise (3 digits)Quadrant bearingENWS60°vN 60° Estart at N, turn60° toward E
Fig 2One vector described three ways: 30 degrees counterclockwise from the positive x-axis, a true bearing of 060 degrees, and a quadrant bearing of N 60 degrees E.
ConventionStart fromTurnRangeExample (same vector)
Standard anglepositive -axis (east)counterclockwise
True bearingnorthclockwise to , written with 3 digits
Quadrant bearingnorth or southtoward east or west to N W
The last column describes one vector pointing up and to the left, written in all three conventions.

To switch between a standard angle and a true bearing , use (or equally ), then add if the result is negative. The two conventions start a quarter-turn apart and turn opposite ways, which is why one formula works both ways. For a quadrant bearing, first decide which quadrant the true bearing falls in: from to it is N E; from to it is S E; from to it is S W; from to it is N W.

NE[N 35° E]start at N, turn 35°NE[E 35° N]start at E, turn 35°These are different directions — 20° apart.First letter = the axis you start from. Last letter = the axis you rotate toward.
Fig 3N 35 degrees E (start at north, turn 35 degrees toward east) and E 35 degrees N (start at east, turn 35 degrees toward north) are different directions, 20 degrees apart.
Read the letters in order
In a quadrant bearing the FIRST letter is where you start and the LAST letter is the way you turn. N W means face north, then turn toward west. Physics courses often write the same direction in square brackets, [N W]. Quadrant bearings in this course start from N or S, so the angle never exceeds .
VideoExplains true bearings and quadrant bearings and how to convert between them, the two compass forms used in this course.Watch on YouTube

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1
A vector has a true bearing of . Write its direction as a quadrant bearing and as a standard angle.
Example 2
A ship sails S E. State its true bearing and its standard angle.
Example 3
is a parallelogram (vertices in order). Name a vector equal to , a vector equal to , and a vector opposite to .

Where marks get lost

  • Treating speed and velocity (or distance and displacement) as the same quantity. Speed is a scalar; velocity is speed together with a direction.
  • Reading N W as from west. The first letter is the starting direction, so it is from north, toward west.
  • Measuring a standard angle clockwise, or a true bearing counterclockwise. Standard angles turn counterclockwise from the positive -axis; bearings turn clockwise from north.
  • Writing for opposite sides of parallelogram . The sides are parallel and equal in length, but the arrows point opposite ways, so .
  • Thinking a negative scalar such as °C has a direction. The sign places it on a scale; it is still a scalar.

Flashcards 7

Scalar
Vector
Equal vectors
Opposite vector
+ 3 more

Practice

1
A hiker is known to be km from Lindsay, Ontario. Why is that not enough to locate the hiker?
2
Which list contains ONLY vector quantities?
3
Match each true bearing to the equivalent quadrant bearing.
  1. A.N E
  2. B.N W
  3. C.S E
  4. D.S W
4
A vector points N W. What is its standard angle (counterclockwise from the positive -axis), in degrees?
°
5
Write the direction S W as a true bearing, in degrees.
°
6
A vector has a standard angle of . What is its true bearing, in degrees?
°
7
Arrange these directions in order of increasing true bearing (clockwise from north).
  1. 1N W
  2. 2S W
  3. 3S E
  4. 4N E
  5. 5true bearing
8
What is the name for the vector that has the same magnitude as but points in exactly the reverse direction? (one word, ___ vector)
9
Two arrows are drawn in different places on a page. Both are cm long and both point N E. Which statement is true?

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