Question Video: Finding the Sum of Vectors in Component Form and Identifying the Resultant on a Grid

Consider two vectors: 𝐀 and 𝐁. 𝐀 = 5𝐒 hat + 1𝐣 hat, and 𝐁 = 2𝐒 hat + 4𝐣 hat. Which of the vectors shown in the diagram is equal to 𝐀 + 𝐁? The grid squares in the diagram have a side length of 1.

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Video Transcript

Consider two vectors 𝐀 and 𝐁. 𝐀 equals five 𝐒 hat plus one 𝐣 hat, and 𝐁 equals two 𝐒 hat plus four 𝐣 hat. Which of the vectors shown in the diagram is equal to 𝐀 plus 𝐁? The grid squares in the diagram have a side length of one.

In the diagram, we’re given a choice of vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓. And we need to say which one of these is equal to 𝐀 plus 𝐁. 𝐒 hat and 𝐣 hat are the unit vectors. And we’re told that the grid squares in the diagram have a side length of one, which means that one 𝐒 hat represents one grid square horizontally and one 𝐣 hat represents one grid square vertically. So one way we can approach this problem is to add the two vectors together in component form.

We do this by adding the components separately. So the horizontal component of 𝐀 plus 𝐁 is equal to five plus two, or seven. And the vertical component is equal to one plus four, which is five. Therefore, 𝐀 plus 𝐁 is equal to seven 𝐒 hat plus five 𝐣 hat. We can now draw 𝐀 plus 𝐁 onto our diagram by counting one, two, three, four, five, six, seven squares in the horizontal direction and then one, two, three, four, five in the vertical direction. And that takes us to the tip of vector 𝐐. And therefore, 𝐀 plus 𝐁 is equal to 𝐐.

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