AP® Physics 1: Algebra-Based · Unit 1 · Topic 1.1
Adding and Subtracting Vectors Along a Line
Vectors along one line add or subtract depending on direction: assign a positive direction, give opposite-pointing vectors a negative sign, then just add the signed numbers to get the resultant.
A vector has size and direction. On a single line there are only two directions to worry about, so you handle direction with a sign instead of an angle. Pick one way as positive, usually the way motion starts or the way most arrows point. Any vector pointing the other way gets a negative number in front of its size. Once every vector has a sign, finding the sum is just arithmetic: add the signed values together.
Laying vectors head to tail makes this visual. Draw the first arrow starting at zero. Start the second arrow where the first one ends. The resultant is the single arrow from the very start to the very end. If both arrows point the same way, the tail of the second sits past the head of the first, and the lengths stack: the resultant is longer than either one. If they point opposite ways, the second arrow backtracks over part of the first, and the resultant is whatever length is left over, pointing in the direction of the longer original arrow. Subtracting a vector is the same process with one extra step: flip the vector you are subtracting 180 degrees, which flips its sign, then add as usual. This is why 5 m/s to the left, when subtracted, acts like 5 m/s to the right.
This signed-number approach is the seed for every later topic in the unit. Position, velocity, and acceleration graphs all rely on treating one direction as positive and the other as negative. Projectile motion later splits motion into perpendicular lines, horizontal and vertical, and adds vectors along each line separately using this exact rule before combining them.
Key terms
- Vector
- A quantity with both size and direction, such as velocity or force.
- Resultant
- The single vector produced by adding two or more vectors together.
- Head to tail
- A method of adding vectors by drawing each one starting where the previous one ended.
- Sign convention
- A rule, like right is positive, used to represent direction along a line with positive or negative numbers.
Watch it explained
- How it worksWalk me through laying two one-dimensional vectors head to tail to get their sum.
- How it worksHow do you subtract one vector from another by flipping it around and adding instead?
- Worked exampleFind the resultant of a 12 m/s arrow pointing right and a 5 m/s arrow pointing left.
- Common mistakeWhy isn't the size of a vector sum just the two sizes added together?
Check yourself
End of topic.