Section 2.2 One-Dimensional Vectors
Key Questions
At the end of this chapter you should be able to answer these questions.
- Given two one-dimensional vectors, how do you compute and then draw the resultant?
- What is the spatial context of multiplying a vector by a scalar?
- How can you find the difference between two one-dimensional vectors, graphically and numerically?
The simplest vector calculations involve one-dimensional vectors. You can learn some important terminology here without much mathematical difficulty.
In one-dimensional problems, all vectors share the same line of action, and either point the same way, or 180° opposite to each other along that line.
Subsection 2.2.1 Vector Addition
Adding multiple vectors together finds the resultant vector, resultant vectors can be thought of as the ‘sum of’ or ‘combination’ of other vectors. To find the resultant vector \(\vec{R}\) of two one-dimensional vectors \(\vec{A}\) and \(\vec{B}\) you can simply use the tip-to-tail technique shown in Figure 2.2.1 below. In the tip-to-tail technique, you slide vector \(\vec{B}\) until its tail is at the tip of \(\vec{A}\text{,}\) and the vector from the tail of \(\vec{A}\) to the tip of \(\vec{B}\) is the resultant \(\vec{R}\text{.}\) Note that the resultant \(\vec{R}\) is the same when you add \(\vec{A}\) onto \(\vec{B}\text{,}\) so the order of vector addition does not matter and is considered commutative.
Subsection 2.2.2 Vector Subtraction
The easiest way to handle vector subtraction is to add the negative of the vector you are subtracting to the other vector. In this way, you can still use the tip-to-tail technique after flipping the vector you are subtracting. You can also simulate this in Figure 2.2.1 by following the steps below.
- Find \(\vec{A}-\vec{B}\) where \(\vec{A}=\cm{2}\) and \(\vec{B}=\cm{2}\text{.}\)
- Set \(\vec{A}\) to a value of \(\cm{2}\) and \(\vec{B}\) to a value of \(\cm{-3}\) (the negative of its actual value).
- Slide the tail of either vector to find the resultant (or sum) of these two vectors. The order does not matter as vector addition is commutative.
- The result should be \(\cm{-1}\text{.}\)
Subsection 2.2.3 Vector Multiplication by a Scalar
Multiplying or dividing a vector by a scalar simply changes the magnitude of the vector but maintains the original line of action. One common transformations is to find the negative of a vector. To find the negative of vector \(\vec{A}\text{,}\) we multiply \(\vec{A}\) by -1, in equation form this looks like:
Spatially, when you create a negative vector, you are simply flipping the tip and tail ends. The magnitude, line of action, and orientation stay the same, but the sense reverses so now the arrowhead points in the opposite direction. This can be simulated in Figure 2.2.1 by changing one of the vectors from a positive to a negative value.