Question Video: Identifying the Relationship Between Vectors Shown on a Grid | Nagwa Question Video: Identifying the Relationship Between Vectors Shown on a Grid | Nagwa

Question Video: Identifying the Relationship Between Vectors Shown on a Grid Physics • First Year of Secondary School

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Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, or 𝐓, shown in the diagram is equal to 𝐀 βˆ’ 𝐁?

01:20

Video Transcript

The diagram shows three vectors: 𝐀, 𝐁, and 𝐂. Which of the following expressions gives 𝐂 in terms of 𝐀 and 𝐁? Is it (A) 𝐂 equals 𝐀 plus 𝐁? (B) 𝐂 equals 𝐀 minus 𝐁. (C) 𝐂 equals 𝐁 minus 𝐀. (D) 𝐂 equals two 𝐀 minus 𝐁. Or (E) 𝐂 equals two 𝐁 minus 𝐀.

To answer this question, we need to work out how to either add or subtract some combination of vectors 𝐀 and 𝐁 to produce vector 𝐂. This is a good opportunity for us to practice adding or subtracting vectors algebraically.

So let’s first recall we can represent a vector as a sum of multiples of unit vectors along the π‘₯- and 𝑦-axes. To see what we mean, let’s look at this given vector 𝐀. Note that it starts at the origin and points left one unit and down six units. Now, as usual, we’ll define the rightward direction as positive along the π‘₯-axis and upward as positive along the 𝑦-axis. So, since the vector 𝐀 points left one unit, we say that it has a horizontal component of negative one unit. Likewise, we say that 𝐀 has a vertical component of negative six units, since we’ve defined downward as negative.

At this point, we should recall that the unit vector along the π‘₯-axis is called 𝐒 hat and that it represents a vector of one unit length in the positive π‘₯-direction. Similarly, the unit vector along the 𝑦-axis is called 𝐣 hat, and it represents a vector of one unit length in the positive 𝑦-direction. So to represent the vector 𝐀 algebraically, we say that 𝐀 equals negative one times 𝐒 hat plus negative six times 𝐣 hat or just negative one 𝐒 hat minus six 𝐣 hat.

Next, let’s look at vector 𝐁. It also starts at the origin, and then points two units upward and five units to the left. Thus, we say that vector 𝐁 equals negative five 𝐒 hat plus two 𝐣 hat.

Okay, now that we’ve defined vectors 𝐀 and 𝐁, let’s start to think about how we could combine them to create vector 𝐂. Recall that when we add or subtract vectors algebraically, we simply add or subtract their like components: 𝐒 hat with 𝐒 hat and 𝐣 hat with 𝐣 hat. For example, let’s see what happens when we add 𝐀 and 𝐁. Adding their horizontal components, negative one 𝐒 hat plus negative five 𝐒 hat gives negative six 𝐒 hat. And adding their vertical components, negative six 𝐣 hat plus two 𝐣 hat gives negative four 𝐣 hat.

So, is this vector 𝐀 plus 𝐁 equal to vector 𝐂, like answer option (A) suggests? To find out, let’s use the diagram to determine the horizontal and vertical components of 𝐂. Note that the tail of the vector is located at this point here, rather than the origin. So, starting here, 𝐂 points four units to the right and eight units downward. This does not correspond to the vector 𝐀 plus 𝐁 that we just found. So we can eliminate option (A). Instead, vector 𝐂 is represented algebraically as four 𝐒 hat minus eight 𝐣 hat.

So now, rather than going through and working out all of the remaining answer options, let’s separately consider the π‘₯- and 𝑦-components of all three of these vectors and think about to combine like components of 𝐀 and 𝐁 to give the respective component of 𝐂. We can start with the vertical components.

Think: how can we combine negative six and two to make negative eight? Well, we know that negative six minus two gives negative eight. So that’s a good hint that vector 𝐀 minus vector 𝐁 gives vector 𝐂. Now, checking the horizontal components, subtracting vector 𝐁 from 𝐀 gives negative one minus negative five, which equals positive four, exactly equal to the horizontal component of vector 𝐂. This confirms that vector 𝐀 minus vector 𝐁 equals vector 𝐂. So we know that answer option (B) is correct. This expression correctly gives vector 𝐂 in terms of vectors 𝐀 and 𝐁.

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