Key Questions
- Why is one-dimensional vector math just like scalar math?
- Given two one-dimensional vectors, how do you compute and then draw the resultant?
- What happens when you multiply a vector by a scalar?
Section 2.2 One-Dimensional Vectors
The simplest vector calculations involve one-dimensional vectors. With one-dimensional vectors, vector arithmetic is the same as ordinary arithmetic.
By definition, one-dimensional vectors all act along a single line. The line they share is called the line of action, but note that vectors that act along a line may point towards either end.
If we place a coordinate axis along the line of action and give vectors that point towards the positive end positive values equal to their magnitudes, and give vectors that point the other way negative values, then the vectors add, subtract and multiply by constants just like ordinary numbers along the number line. The resulting sign indicates the direction of the result. Note that, by convention, the labeled end of an axis is the positive end.
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 two or more vectors.
To find the resultant vector \(\vec{R}\) of two one-dimensional vectors \(\vec{A}\) and \(\vec{B}\) you can use the tip-to-tail technique in Figure 2.1.1 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 vector addition is commutative: the resultant \(\vec{R}\) is the same whether you add \(\vec{A}\) to \(\vec{B}\) or \(\vec{B}\) to \(\vec{A}\text{.}\)
Instructions.
The interactive shows \(\vec{A}+ \vec{B}= \vec{R}.\)
You can adjust the magnitude and direction of vectors with the tips and their position along the line of action with the tails. When they arranged tip-to-tail, the resultant vector will appear. The vectors are represented as scalar components multiplied by unit vector \(\ihat\text{.}\)
This is just a graphic representation of tip-to-tail addition; \(2\ihat + 3 \ihat = 5 \ihat\) regardless of where \(\vec{A}\) and \(\vec{B}\) are located on the line of action.
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.
\begin{equation}
\vec{A} - \vec{B} = \vec{A} + (-\vec{B})\tag{2.2.1}
\end{equation}
Example 2.2.2. Vector subtraction.
Find \(\vec{A}-\vec{B}\) where \(\vec{A}=2\ \ihat\) and \(\vec{B}=3\ \ihat\text{.}\)
Answer.
\begin{equation*}
\vec{R}= -1\ \ihat\text{.}
\end{equation*}
Solution.
You can simulate this in Figure 2.2.1.
- Set \(\vec{A}\) to a value of \(2\ \ihat\) and \(\vec{B}\) to a value of \(-3\ \ihat\text{,}\) the negative of its actual value.
- Move the vectors until they are tip-to-tail. The order does not matter because vector addition is commutative.
\begin{equation*}
\vec{R}= -1\ \ihat\text{.}
\end{equation*}
Subsection 2.2.3 Vector Multiplication by a Scalar
Multiplying or dividing a vector by a scalar changes the vector’s magnitude but maintains its original line of action. In other words, multiplying a vector by a scalar "scales" the vector.
One common transformation is to find the negative of a vector. To find the negative of vector \(\vec{A}\text{,}\) we multiply it by -1; in equation form
\begin{equation*}
-\vec{A} =(-1) \vec{A}
\end{equation*}
Spatially, the effect of multiplying a vector by \(-1\text{,}\) is to reverse it end for end. The magnitude and line of action stay the same, but the sense reverses, so now the arrowhead points in the opposite direction. When multiplying vector components by \(-1\text{,}\) simply flip the +/- sign of each term to flip the direction of the vector.
