11.3 Vector addition and subtraction: analytical methods

Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made. Analytical methods are limited only by the accuracy and precision with which physical quantities are known.

Resolving a vector into perpendicular components

Analytical techniques and right triangles go hand-in-hand in physics because (among other things) motions along perpendicular directions are independent. We very often need to separate a vector into perpendicular components. For example, given a vector like $\mathbf{A}$ in [link] , we may wish to find which two perpendicular vectors, ${\mathbf{A}}_x$ and ${\mathbf{A}}_y$ , add to produce it.

${\mathbf{A}}_x$ and ${\mathbf{A}}_y$ are defined to be the components of $\mathbf{A}$ along the x - and y -axes. The three vectors $\mathbf{A}$ , ${\mathbf{A}}_x$ , and ${\mathbf{A}}_y$ form a right triangle:

Note that this relationship between vector components and the resultant vector holds only for vector quantities (which include both magnitude and direction). The relationship does not apply for the magnitudes alone. For example, if ${\mathbf{\text{A}}}_x=\mathrm{3\; m}$ east, ${\mathbf{\text{A}}}_y=\mathrm{4\; m}$ north, and $\mathbf{\text{A}}=\mathrm{5\; m}$ north-east, then it is true that the vectors ${\mathbf{A}}_x{\text{ + A}}_y\text{ = A}$ . However, it is not true that the sum of the magnitudes of the vectors is also equal. That is,

Thus,

If the vector $\mathbf{A}$ is known, then its magnitude $A$ (its length) and its angle $\theta$ (its direction) are known. To find $A_x$ and $A_y$ , its x - and y -components, we use the following relationships for a right triangle.

and

Suppose, for example, that $\mathbf{A}$ is the vector representing the total displacement of the person walking in a city considered in Kinematics in Two Dimensions: An Introduction and Vector Addition and Subtraction: Graphical Methods .

Then $A=10.3$ blocks and $\theta =\mathrm{29.1º}$ , so that

Calculating a resultant vector

If the perpendicular components ${\mathbf{A}}_x$ and ${\mathbf{A}}_y$ of a vector $\mathbf{A}$ are known, then $\mathbf{A}$ can also be found analytically. To find the magnitude $A$ and direction $\theta$ of a vector from its perpendicular components ${\mathbf{A}}_x$ and ${\mathbf{A}}_y$ , we use the following relationships: