Question Video: Finding an Equivalent Vector Graphically | Nagwa Question Video: Finding an Equivalent Vector Graphically | Nagwa

# Question Video: Finding an Equivalent Vector Graphically Mathematics • First Year of Secondary School

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Which of the following is equivalent to 1/2(๐ถ๐ต โ ๐ท๐ถ)? [A] ๐ด๐ธ [B] ๐ต๐ธ [C] ๐ถ๐ธ [D] ๐ถ๐ด [E] ๐ธ๐ถ

05:49

### Video Transcript

Which of the following is equivalent to one-half times the vector from ๐ถ to ๐ต minus the vector from ๐ท to ๐ถ? Is it option (A) the vector from ๐ด to ๐ธ? Option (B) the vector from ๐ต to ๐ธ. Option (C) the vector from ๐ถ to ๐ธ. Option (D) the vector from ๐ถ to ๐ด. Or is it option (E) the vector from ๐ธ to ๐ถ?

In this question, we need to determine which of five given vectors is equivalent to a given expression involving vectors. And we want to do this by using the given diagram. Letโs start by looking at the given vector expression weโre given. This vector expression is in two parts, first when multiplying an entire expression by one half, and inside of our parentheses, we can see we have the difference between two vectors. Letโs start by looking at the inner parentheses since this is the first operation carried out, vector from ๐ถ to ๐ต minus vector from ๐ท to ๐ถ.

To find the difference of two vectors, we can do this by using the diagram. However, letโs start by sketching these two vectors on our given diagram. Letโs start with the vector from ๐ถ to ๐ต. Its initial point will be point ๐ถ and its terminal point will be vector ๐ต, so we draw an arrow from ๐ถ to ๐ต as shown. And itโs worth recalling here vectors are defined entirely in terms of their magnitude and direction. So only the direction of this arrow and the length of this arrow define the vector.

So, for example, by using our knowledge of parallelograms, we could draw the vector from ๐ถ to ๐ต starting at point ๐ท and ending at point ๐ด. We know that these two vectors have the same direction since ๐ด๐ต๐ถ๐ท is a parallelogram, so the side ๐ด๐ท is parallel to the side ๐ต๐ถ. Similarly, we know they have the same magnitude since opposite sides in a parallelogram have the same length. Letโs follow the same process to sketch the vector from ๐ท to ๐ถ. The initial point of this vector is vertex ๐ท and the terminal point of this vector is vertex ๐ถ. This gives us the following vector from ๐ท to ๐ถ. We could follow the same reasoning to determine the vector from ๐ท to ๐ถ is equivalent to the vector from ๐ด to ๐ต.

However, before we do this, letโs look at the expression we need to evaluate. Inside our parentheses, weโre subtracting the vector from ๐ท to ๐ถ from the vector from ๐ถ to ๐ต. And thereโs several different ways we can evaluate this expression. However, the easiest way is to notice that weโre subtracting one vector from another, and itโs usually easier to add vectors together. And we can recall multiplying a vector by negative one switches its direction. However, it leaves its magnitude unchanged. So in this case, negative one times the vector from ๐ท to ๐ถ will be equal to the vector from ๐ถ to ๐ท. And of course, this result holds true in general. Negative one times a vector from a point ๐ to a point ๐ will just be the vector from ๐ to ๐. Therefore, we have that the vector from ๐ถ to ๐ต minus the vector from ๐ท to ๐ถ is equal to the vector from ๐ถ to ๐ต added to the vector from ๐ถ to ๐ท.

We can also sketch this on our diagram by switching the direction of our vector. This then gives us the following. We now want to evaluate the vector from ๐ถ to ๐ต added to the vector from ๐ถ to ๐ท. We can do this by recalling how we add two vectors together graphically. This is sometimes called the triangle rule for vector addition. If we have two vectors ๐ฎ and ๐ฏ drawn tip to tail so the terminal point of vector ๐ฎ is the initial point of vector ๐ฏ, then the vector ๐ฎ added to the vector ๐ฏ is the vector which starts at the initial point of vector ๐ฎ and finishes at the terminal point of vector ๐ฏ.

Another way of thinking about this is we can add two vectors drawn in this way by following the arrows since weโre just adding their displacements. We want to use this to add the vector from ๐ถ to ๐ต to the vector from ๐ถ to ๐ท. And we can do this by using our diagram and our knowledge of vector addition.

There are a few different ways of doing this. However, weโll only go through one of these. Letโs start by looking at the vector from ๐ถ to ๐ต. It starts at vertex ๐ถ and ends at vertex ๐ต. So to add this to the vector from ๐ถ to ๐ท by using our rule, we need the vector from ๐ถ to ๐ท to have its initial point at vector ๐. And we can do this by using one of our previous results. Opposite sides in a parallelogram have the same direction since theyโre parallel and they have the same magnitude. This means the vector from ๐ต to ๐ด must be equal to the vector from ๐ถ to ๐ท since ๐ด๐ต๐ถ๐ท is a parallelogram.

And now, we can easily add these two vectors together by noticing the terminal point of the vector from ๐ถ to ๐ต is coincident with the initial point of the vector from ๐ต to ๐ด, which is the same as the vector from ๐ถ to ๐ท. This means the vector from ๐ถ to ๐ต added to the vector from ๐ถ to ๐ท will be the vector with initial point at ๐ถ and terminal point at ๐ด. However, weโre not done yet. Remember, weโre only working with the expression inside of the parentheses. So this is only the vector ๐ถ๐ต minus the vector ๐ท๐ถ.

We still need to multiply this vector by one-half. And we can do this by recalling multiplying a vector by a positive scalar does not affect its direction; it only affects its magnitude. In particular, multiplying a vector by one-half halves its magnitude. So we need to find a vector in the same direction as this vector. However, it needs to have half the magnitude. And we can do this by noting weโre given the bisector of the line segment from ๐ถ to ๐ด. Itโs point ๐ธ. Therefore, the vector from ๐ธ to ๐ด and the vector from ๐ถ to ๐ธ will be half the magnitude of this vector in the same direction. And of these two options, we can see that only the vector from ๐ถ to ๐ธ appears in the five given options.

Therefore, we were able to show of the five given options, only option (C) the vector from ๐ถ to ๐ธ is equivalent to a half times the vector from ๐ถ to ๐ต minus the vector from ๐ท to ๐ถ.

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