Question 3.1: Adding Vectors Head to Tail Three vectors A, B and C are sho......
Adding Vectors Head to Tail
Three vectors \overrightarrow{ A }, \overrightarrow{ B }, \text { and } \overrightarrow{ C } are shown in Figure 3.8. Find the vector \overrightarrow{ R }=\overrightarrow{ A }+\frac{1}{2} \overrightarrow{ B }-2 \overrightarrow{ C }
INTERPRET and ANTICIPATE
As discussed, there are several ways to add and subtract vectors. Because each method produces the same result, the choice is a matter of personal taste. In this example, we choose the head-to-tail method.
SOLVE
Regardless of which addition method we choose, our first step must be to multiply vectors \overrightarrow{ B } and \overrightarrow{ C } by the appropriate scalars. In Figure 3.9, \frac{1}{2} \overrightarrow{ B } points in the same direction as \overrightarrow{ B } but is only half as long. Similarly, -2\overrightarrow{ C } points in the direction opposite that of \overrightarrow{ C } and is twice as long as \overrightarrow{ C } .
We arbitrarily choose to add by the head-to-tail method. First, add \overrightarrow{ A }+\frac{1}{2} \overrightarrow{ B } by placing the tail of \frac{1}{2} \overrightarrow{ B } next to the head of \overrightarrow{ A } (Fig. 3.10). In this problem, the vectors \overrightarrow{ A },\frac{1}{2} \overrightarrow{ B } and \left(\overrightarrow{ A }+\frac{1}{2} \overrightarrow{ B }\right) all point in the same direction.
Finally, we use head-to-tail addition to combine -2 \overrightarrow{ C } and \left(\overrightarrow{ A }+\frac{1}{2} \overrightarrow{ B }\right) as shown in Fig. 3.11.
CHECK and THINK
The resultant vector points to the left and downward. As a check, you can combine the vectors in another order. For example, you could first combine \frac{1}{2} \overrightarrow{ B }-2 \overrightarrow{ C } and then add \overrightarrow{ A }. Try it. The resultant should be the same, verifying that vector addition is associative.
