Key takeaway: A vector is an arrow-shaped quantity that records both how much and which way, written in components as ⟨a, b⟩ for a horizontal change a and a vertical change b.
Grade 12 – PrecalculusN-VM.A.1Geometry and Measurement
Definition of Vector
A vector captures a movement or a push: how far and in which direction. A wind of 14 mph from the northwest qualifies; so does a step of 3 blocks east and 2 blocks north. A plain number such as 14 or 5 carries only a size and is called a scalar.
Drawn on a plane, it is an arrow whose length shows the size and whose pointing direction shows the heading. Where the arrow starts does not matter: two arrows with the same length and heading are equal. In component form each arrow becomes a pair giving the horizontal and vertical change.
Formula and notation
Travel from point \(P(x_1,y_1)\) to point \(Q(x_2,y_2)\) and the components are
\[\vec{PQ}=\langle x_2-x_1,\ y_2-y_1\rangle.\]
Printed names are bold, as in \(\mathbf{v}\), or carry an arrow, as in \(\vec{v}\). Adding two of these means adding matching components, and scaling means multiplying each component.
Key facts
- Equal components mean equal arrows, wherever they are drawn.
- The zero arrow ⟨0, 0⟩ has no length and no direction.
- Reversing a vector negates both components, and a vector can be moved without changing.
- Velocity, force and displacement have direction; time, mass and temperature do not.
Vector example
A scout walks from camp at \(A(-2,5)\) to a lookout at \(B(4,-3)\) on a map grid measured in blocks. Write the displacement as components and find how far the scout travels in a straight line.
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Subtract start from end: east-west change \(4-(-2)=6\), north-south change \(-3-5=-8\).
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Straight-line distance is the arrow’s length, from the Pythagorean theorem: root of \(36+64\), which is \(\sqrt{100}\).
The displacement is \(\langle 6,-8\rangle\), a straight-line trip of 10 blocks.
Common mistake
Watch out
Camila finds the trip from A to B as \(A-B=\langle -6,8\rangle\), subtracting the end point from the start point.
Why it is wrong
The terminal point goes first in the subtraction: end minus start. She reversed the order, so her arrow points from B back to A. The correct vector for A to B has components \(\langle 6,-8\rangle\).
Related terms and lessons
Browse the glossary: A–Z list · Precalculus vocabulary · Geometry and Measurement