11.2 Vector addition and subtraction: graphical methods

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• Understand the rules of vector addition, subtraction, and multiplication.
• Apply graphical methods of vector addition and subtraction to determine the displacement of moving objects.

Vectors in two dimensions

A vector    is a quantity that has magnitude and direction. Displacement, velocity, acceleration, and force, for example, are all vectors. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. In two dimensions (2-d), however, we specify the direction of a vector relative to some reference frame (i.e., coordinate system), using an arrow having length proportional to the vector’s magnitude and pointing in the direction of the vector.

[link] shows such a graphical representation of a vector , using as an example the total displacement for the person walking in a city considered in Kinematics in Two Dimensions: An Introduction . We shall use the notation that a boldface symbol, such as $\text{D}$ , stands for a vector. Its magnitude is represented by the symbol in italics, $D$ , and its direction by $\theta$ .

Vectors in this text

In this text, we will represent a vector with a boldface variable. For example, we will represent the quantity force with the vector $\text{F}$ , which has both magnitude and direction. The magnitude of the vector will be represented by a variable in italics, such as $F$ , and the direction of the variable will be given by an angle $\theta$ .

The head-to-tail method    is a graphical way to add vectors, described in [link] below and in the steps following. The tail    of the vector is the starting point of the vector, and the head (or tip) of a vector is the final, pointed end of the arrow.

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