Definition: A vector is a quantity with both magnitude and direction, drawn as an arrow and written in components such as the ordered pair with horizontal and vertical parts.
What is a vector?
A vector records a movement: how far and in which direction. In the plane we write it by components, <a, b>, meaning a units horizontally and b units vertically. Two vectors are equal when they have the same components, wherever they are drawn.
How do you combine vectors?
- Add component by component: <a, b> + <c, d> = <a + c, b + d>.
- Scale by a number k: k<a, b> = <ka, kb>.
Worked example: let u = <3, -4> and v = <5, 2>. Then u + v = <8, -2> and 2u – v = <6 – 5, -8 – 2> = <1, -10>.
Real-life picture: a boat heads east at 5 mph while the current pushes it north at 2 mph. The two velocity vectors add to give the actual motion.
What is a common mistake?
Adding magnitudes instead of components. Two vectors of lengths 3 and 4 do not necessarily sum to length 7. Mathbot adds components first, then measures.