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Section 2.2 One-Dimensional Vectors

The simplest vector calculations involve one-dimensional vectors. You can learn some important terminology here without much mathematical difficulty. In one-dimensional situations, all vectors share the same line of action, but may point towards either end. If the line of action has a positive end like a coordinate axis does, then a vector pointing towards that end will have a positive scalar component.

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 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.

Figure 2.2.1. One Dimensional Vector Addition

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.

  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.

  2. 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. One common transformations 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 negating a vector this way is to rotate it by 180°. The magnitude, line of action, and orientation stay the same, but the sense reverses so now the arrowhead points in the opposite direction.