What Vectors Are

The Treasure Walk began with an instruction that was almost enough: walk 20 metres. Twenty metres which way? Every group that guessed a direction ended up somewhere different, and the treasure stayed buried. The fix was an instruction with two parts — 20 metres, bearing N 40° W — and that two-part quantity is a vector: a magnitude and a direction, welded together. A plain number with no direction attached (the 20 alone, a temperature, a mass) is called a scalar by contrast.

The world is full of quantities that are secretly vectors: a displacement, a force on a bridge member, the velocity your phone’s GPS reports, the nudge a game engine gives a sprite. Whenever “how much” is not enough without “which way”, a vector is underneath.

Self-check: a friend is 69 km from Lindsay, Ontario.

Where are they? (click to expand) You cannot say — and that is the point. Sixty-nine kilometres is only a magnitude; without a direction it describes an entire circle of possible positions around Lindsay. Position needs a vector: 69 km on a stated bearing pins your friend to one spot.

Arrows, honestly drawn

On paper a vector is a directed line segment — an arrow. Its length is the magnitude, written $|\vec{v}|$; its direction can be given as a rotation like $320°$ or a bearing like N 40° W. And here is the convention that makes the algebra work: an arrow slid to a new position without turning or stretching is still the same vector. Vectors have magnitude and direction, but no fixed address.

Components: arrows as coordinates

Geometry draws vectors; algebra prefers to file them. Put the tail at the origin and record where the tip lands: the vector with magnitude 10 pointing $36.9°$ above the positive $x$-axis files as $(8,6)$, because $10\cos 36.9°\approx 8$ and $10\sin 36.9°\approx 6$. The translation runs both ways — from $(8,6)$, the Pythagorean theorem recovers $|\vec{v}|=\sqrt{8^2+6^2}=10$ and ${\tan }^{-1}\left(\frac{6}{8}\right)$ recovers the direction.

Best of all, nothing stops at two dimensions. A vector in three-space is a triple like $(1,2,2)$, with magnitude $\sqrt{1+4+4}=3$ by the same theorem used twice — and suddenly you can do geometry in a space you cannot fully draw. That is the quiet superpower of the whole unit: components turn pictures into arithmetic, and arithmetic works in any number of dimensions.

Adding and Scaling Vectors is what these arrows can do; the warm-up questions in Dot and Cross Product Practice rehearse the translations both ways.

Curriculum connection

C1.1

recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement, forces involved in structural design, simple animation of computer graphics, velocity determined using GPS) Sample problem: Position is represented using vectors. Explain why knowing that someone is 69 km from Lindsay, Ontario, is not sufficient to identify their exact position.

Link to original

C1.2

represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways (e.g., $320^{\circ }$; N 40° W), and algebraically (e.g., using Cartesian coordinates; using polar coordinates), and recognize vectors with the same magnitude and direction but different positions as equal vectors

Link to original

C1.3

determine, using trigonometric relationships [e.g., $x=r\cos \theta$, $y=r\sin \theta$, $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$ or ${\tan }^{-1}\left(\frac{y}{x}\right)+180^{\circ }$, $r=\sqrt{x^2+y^2}$], the Cartesian representation of a vector in two-space given as a directed line segment, or the representation as a directed line segment of a vector in two-space given in Cartesian form [e.g., representing the vector $(8,6)$ as a directed line segment] Sample problem: Represent the vector with a magnitude of 8 and a direction of $30^{\circ }$ anticlockwise to the positive x-axis in Cartesian form.

Link to original

C1.4

recognize that points and vectors in three-space can both be represented using Cartesian coordinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations

Link to original