2.3 Vector addition and subtraction: analytical methods  (Page 3/6)

Components along the same axis, say the x -axis, are vectors along the same line and, thus, can be added to one another like ordinary numbers. The same is true for components along the y -axis. (For example, a 9-block eastward walk could be taken in two legs, the first 3 blocks east and the second 6 blocks east, for a total of 9, because they are along the same direction.) So resolving vectors into components along common axes makes it easier to add them. Now that the components of $\mathbf{R}$ are known, its magnitude and direction can be found.

Step 3. To get the magnitude $R$ of the resultant, use the Pythagorean theorem:

Step 4. To get the direction of the resultant:

The following example illustrates this technique for adding vectors using perpendicular components.

Adding vectors using analytical methods

Add the vector $\mathbf{A}$ to the vector $\mathbf{B}$ shown in [link] , using perpendicular components along the x - and y -axes. The x - and y -axes are along the east–west and north–south directions, respectively. Vector $\mathbf{A}$ represents the first leg of a walk in which a person walks $\text{53}\text{.}\text{0 m}$ in a direction $\text{20}\text{.}0\text{º}$ north of east. Vector $\mathbf{B}$ represents the second leg, a displacement of $\text{34}\text{.}\text{0 m}$ in a direction $\text{63}\text{.}0\text{º}$ north of east.

Strategy

The components of $\mathbf{A}$ and $\mathbf{B}$ along the x - and y -axes represent walking due east and due north to get to the same ending point. Once found, they are combined to produce the resultant.

Solution

Following the method outlined above, we first find the components of $\mathbf{A}$ and $\mathbf{B}$ along the x - and y -axes. Note that $A=\mathrm{53.0\; m}$ , ${\theta }_{\text{A}}=\mathrm{20.0º}$ , $B=\mathrm{34.0\; m}$ , and ${\theta }_{\text{B}}=\mathrm{63.0º}$ . We find the x -components by using $A_x=A\text{cos}\theta$ , which gives

and

Similarly, the y -components are found using $A_y=A\text{sin}{\theta }_{\mathrm{A}}$ :

and

The x - and y -components of the resultant are thus

and

Now we can find the magnitude of the resultant by using the Pythagorean theorem:

so that

Finally, we find the direction of the resultant:

Thus,

Discussion

This example illustrates the addition of vectors using perpendicular components. Vector subtraction using perpendicular components is very similar—it is just the addition of a negative vector.

Subtraction of vectors is accomplished by the addition of a negative vector. That is, $\mathbf{A}-\mathbf{B}\equiv \mathbf{A}+(\mathbf{–B})$ . Thus, the method for the subtraction of vectors using perpendicular components is identical to that for addition . The components of $\mathbf{\text{–B}}$ are the negatives of the components of $\mathbf{B}$ . The x - and y -components of the resultant $\mathbf{A}-\text{B = R}$ are thus