Vector
In short: A vector bundles a length with a heading: it is pictured as an arrow and recorded as components, so ⟨8, -4⟩ means a move of 8 units right and 4 units down.
PrecalculusGeometryN-VM.A.1
What is Vector?
A vector describes a movement or a push, not just an amount. “20 miles” is only a distance, but “20 miles northeast” is a vector, because it carries direction as well.
In the plane, a vector from an initial point \(P(x_1, y_1)\) to a terminal point \(Q(x_2, y_2)\) has component form \(\langle x_2 – x_1,\ y_2 – y_1\rangle\). Two arrows with the same components are the same vector, wherever they are drawn. Adding vectors adds components, and multiplying by a scalar \(k\) multiplies each component by \(k\).
How to spot it
- A story mentions wind, current, force or velocity with both a speed and a heading.
- Two points are given and you must describe the trip from one to the other with a vector.
- Arrows are drawn on a grid and combined head to tail, or scaled to be twice as long.
- Notation such as ⟨3, -2⟩ or an arrow over a letter, like v, appears.
The method
- Subtract the initial point’s coordinates from the terminal point’s coordinates, x with x and y with y.
- Write the two differences in angle brackets: that is the component form.
- To scale, multiply each component by the scalar; to add, add components in matching order.
- Draw the arrow from the initial point to check that it points the way the numbers say.
Vector: solved example
Problem
On a map grid, a survey drone travels from \(P(-3, 6)\) to \(Q(5, 2)\). Record that trip as components, then record a trip covering only half the distance in the same direction.
Solution
Subtract coordinates: \(\overrightarrow{PQ} = \langle 5 – (-3),\ 2 – 6\rangle = \langle 8, -4\rangle\). That is 8 units right and 4 units down.
Half of it multiplies each component by \(\frac{1}{2}\): \(\langle 4, -2\rangle\), which points the same way but is half as long.
Try it
Question 1
Write the component form of the vector from \((4, -1)\) to \((-2, 8)\).
Show the answer
\(\langle -2 – 4,\ 8 – (-1)\rangle = \langle -6, 9\rangle\).
Question 2
Add \(\langle 3, -2\rangle\) and \(\langle -7, 5\rangle\).
Show the answer
\(\langle 3 + (-7),\ -2 + 5\rangle = \langle -4, 3\rangle\).
Vector vs. Magnitude of a Vector
This is the whole object: direction and size together. Its magnitude is only the size, one nonnegative number. Vectors \(\langle 3, 4\rangle\) and \(\langle -3, -4\rangle\) point opposite ways yet share the magnitude 5.
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Related words
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