Video: Choosing the Resultant of a Sum of Two Vectors

Some vectors are drawn to scale on a square grid. Which of the vectors I, II, III, or IV is the resultant of the vectors 𝐀 and 𝐁?

02:40

Video Transcript

Some vectors are drawn to scale on a square grid. Which of the vectors 1, 2, 3, or 4 is the resultant of the vectors 𝐀 and 𝐁?

So here we have the square grid with a number of vectors drawn in. Here’s vector 1, here’s vector 2, here’s number 3, and here’s number 4. These are all candidates for the vector that results from adding vector 𝐀, shown here, with vector 𝐁, shown here. So the question is, if we add together 𝐀 and 𝐁, then which of these four vectors do we get?

Now looking at this grid, we can see that vector 𝐀 is a completely vertical vector. That is, it has no horizontal component. And likewise, vector 𝐁 is a purely horizontal vector with no vertical component. If we add these two vectors together, we can see that the resultant will have both a vertical and a horizontal component to it. It must because when we combine 𝐀 and 𝐁, we have both of those components.

This eliminates a few of our answer options. We can see, for example, that vector 1 only has a vertical and no horizontal component. Therefore, it won’t be our answer. And then down at the bottom of our grid, we have vector 4, which has only a horizontal and no vertical component. Therefore, we know this won’t be our choice either.

Now we want to figure out whether to identify vector 3 or vector 2 as the resultant of vectors 𝐀 and 𝐁. To do that, we’ll use the grid spacings marked out on the square grid. Starting at the origin, where the tails of vector 𝐀 and vector 𝐁 meet. What we’ll do is we’ll count grid spaces from this point until we get to the head of vector 𝐀 and vector 𝐁, respectively. Let’s start by moving horizontally out to the tip of vector 𝐁.

Beginning at the origin, we count one grid space, two grid spaces, three, four, and then five. This means that our resultant vector, the vector that comes from adding vector 𝐀 and vector 𝐁, will also have five horizontal spaces to it. And then doing the same thing on the vertical axis towards the tip of vector 𝐀, we once again start at the origin and then count up grid spaces. One, two, three, four, and again five.

So the resultant of vectors 𝐀 and 𝐁 will have five grid spaces in the horizontal direction and five in the vertical direction. And of our two remaining candidates, vectors 3 and 2, we can see that it’s vector 3 that meets this test. Starting at the origin, if we move out one, two, three, four, five spaces in the horizontal direction. And then one, two, three, four, five spaces in the vertical direction. We reach this point here, which is at the tip of vector 3. And we see that the tail of vector 3 is at the origin. So vector 3 is the one which is the resultant of the vectors 𝐀 and 𝐁. Vector 3 then is the answer we choose.

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