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 π, π, π, π, or π, shown in the diagram is equal to π β π?

01:20

### Video Transcript

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

To answer this question, we need to work out how to either add or subtract some combination of vectors π and π to produce vector π. This is a good opportunity for us to practice adding or subtracting vectors algebraically.

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

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

Next, letβs look at vector π. It also starts at the origin, and then points two units upward and five units to the left. Thus, we say that vector π equals negative five π’ hat plus two π£ hat.

Okay, now that weβve defined vectors π and π, letβs start to think about how we could combine them to create vector π. Recall that when we add or subtract vectors algebraically, we simply add or subtract their like components: π’ hat with π’ hat and π£ hat with π£ hat. For example, letβs see what happens when we add π and π. Adding their horizontal components, negative one π’ hat plus negative five π’ hat gives negative six π’ hat. And adding their vertical components, negative six π£ hat plus two π£ hat gives negative four π£ hat.

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

So now, rather than going through and working out all of the remaining answer options, letβs separately consider the π₯- and π¦-components of all three of these vectors and think about to combine like components of π and π to give the respective component of π. 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 thatβs a good hint that vector π minus vector π gives vector π. Now, checking the horizontal components, subtracting vector π from π gives negative one minus negative five, which equals positive four, exactly equal to the horizontal component of vector π. This confirms that vector π minus vector π equals vector π. So we know that answer option (B) is correct. This expression correctly gives vector π in terms of vectors π and π.

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