Video: Vector Resultants

In this video we learn how to combine vectors graphically and algebraically, finding the resultant that shows the net effect of adding a group of vectors together.

08:17

Video Transcript

In this video, we’re going to learn about vector resultants, what they are, how they’re useful, and how to calculate them practically.

To get an idea for vector resultants, imagine that you are the captain of a pirate ship. And you’ve received reports of buried treasure on an island you’re now approaching in your ship. You’re concerned that someone else might find the buried treasure before you do. So, you want to minimize the time it will take to get to that spot on your map and start digging.

Between your current location and the island you’re going to you, there’s a strong cross current of 0.5 meters per second. Through measurements of your own ship’s speed, you know that you’re moving at three meters per second with the wind. The question is, what distance do you want to aim above the treasure on the shoreline so that when you factor in the effects of the cross current, your path will lead you directly towards the X on the map? To answer this question, we need to understand vector resultants.

When we talk about vector resultants, we’re speaking of the overall, or net, vector that’s created by the combination of two or more vectors. There are two ways, in general, that we can combine vectors. The first way is by combining them graphically. For example, if we had vectors 𝐴, 𝐵, and 𝐶 drawn out and put them tip to tail, then we could find the resultant vector by connecting the tail of vector 𝐴 with the tip of vector 𝐶. So that’s the first method for finding a vector resultant.

The second method is by combining the vectors algebraically. In this method, our vectors, say we had 𝐴, 𝐵, and 𝐶 once more, would be determined by their component parts numerically. We would then add these three vectors together by components to solve for our resultant 𝑅.

Resultant vectors are useful because they help us understand net overall effects in a system. Or, as in our first example, they help us solve for the shortest possible path between two different points. Let’s get some practice with these two methods of combining vectors to find a resultant.

Imagine that you’re competing in the event of geocaching where, based on instructions and clues that you find at certain geographical locations, you’re guided to an end point where you’ll find the prize. As you approach the end of the race, you discover at your current location that there are only three clues remaining between you and the end point. Taking out your map, which is marked out with grid points every one meter, looking at the last three instructions, and understanding that time is of the essence, you decide to mark out each step on your map and see if there’s a shorter way to go about getting to the end point.

You map out the first instruction to go six meters north from your current location. From there, you’re told to travel east, a distance of eight meters. You mark that out on your map as well. Then, lastly, you mark out the third leg of this final stage of your journey. This must be where the prize is located. By graphically combining these three vectors, based on the instructions for this race, you see that you can walk a significantly shorter distance to get from where you are to the prize.

By graphically resolving these three vectors and finding the resultant, you’ve increased your chances of getting to the prize quickly. Now let’s look at a vector resultant example where we combine vectors algebraically.

Three displacement vectors 𝐀, 𝐁, and 𝐂 have magnitudes 10.0 meters, 7.0 meters, and 8.0 meters, respectively. The directions of 𝐀, 𝐁, and 𝐂 make counterclockwise angles of 35 degrees, negative 110 degrees, and 30 degrees, respectively, with the positive 𝑥-axis, as shown in the diagram. Calculate 𝐀 plus 𝐁 plus 𝐂. Calculate 𝐀 minus 𝐁. Calculate 𝐀 minus three 𝐁 plus 𝐂.

This exercise involves calculating three vector resultants. The first is 𝐀 plus 𝐁 plus 𝐂. Then, we wanna calculate 𝐀 minus 𝐁. And finally, 𝐀 minus three 𝐁 plus 𝐂. We’ll use the magnitudes of each vector as well as the directions that they point in order to write them in terms of their components to combine them. Let’s do that now, beginning with vector 𝐀.

In this diagram showing our three vectors 𝐀, 𝐁, and 𝐂, we can define direction to the right as direction in the positive 𝑖-direction, or positive 𝑥, and up as movement in the positive 𝑗-direction. Based on that convention, we can write the components of our vector 𝐀 as the magnitude of 𝐀, 10.0 meters, times the cos of 35 degrees in the 𝑖-direction plus the sin of 35 degrees in the 𝑗-direction. For the vector 𝐁, we can write that as the magnitude of 𝐁, 7.0 meters, times the cos of 110 degrees in the 𝑖-direction minus the sin of 110 degrees in the 𝑗-direction. And finally, for vector 𝐂, that’s equal to the magnitude of 𝐂, 8.0 meters, times the cos of 30 degrees in the 𝑖-direction plus the sin of 30 degrees in the 𝑗-direction.

To solve for the first vector resultant we wanna find, 𝐀 plus 𝐁 plus 𝐂, we’ll add these three vectors by their component parts. That is, we’ll add up the 𝑖-components of 𝐀, 𝐁, and 𝐂 and, separately, the 𝑗-components of 𝐀, 𝐁, and 𝐂. And when we do, we find the result of 12.7𝑖 plus 3.2𝑗 meters. This is the resulting vector from adding 𝐀, 𝐁, and 𝐂.

Next, we move on to finding the difference of 𝐀 minus 𝐁. Once again, we’ll use our vectors in their component form, 𝐀 and 𝐁. But this time, we’ll subtract 𝐁 from 𝐀 instead of adding them together. Once again, we’ll be careful to keep the vectors separated out by their 𝑖- and 𝑗-components. Entering these values on our calculator, we find that this difference is 10.6𝑖 plus 12.3𝑗 meters. That’s the vector resultant of subtracting vector 𝐁 from vector 𝐀.

Finally, we want to solve for 𝐀 minus three times 𝐁 plus 𝐂. To calculate this resultant, we can multiply vector 𝐁, both sides, by negative three, giving us a value of negative 21.0 meters as its new magnitude. We’re now ready to add these three vectors together by their components. When we do, we find a result of 22.3𝑖 plus 29.5𝑗 meters. That’s the resultant of adding vectors 𝐀 and 𝐂 together and subtracting three times vector 𝐁.

So, we’ve now had some experience in finding vector resultants, both graphically and algebraically. To summarize this topic, vector resultants show the net effect of a combination of vectors. Resultants can be found either graphically or algebraically.

And if we recall back to our opening question of what point along the shoreline do we aim our ship for in order to arrive directly at the treasure, including the cross current. If the distance between our ship and the shoreline is 𝐿, then given the speed of our boat towards the shore and the speed of the cross current affecting our travel. We’d want to aim a distance 𝑑 of 𝐿 over six meters above the treasure on the shoreline. That would get us there in the shortest possible path. Finding vector resultants is both a common and a useful practice in solving various physics exercises.

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