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.
