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Question Video: Expressing a Vector in Component Form Physics • First Year of Secondary School

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Write 𝐀 in component form.

01:36

Video Transcript

Write 𝐀 in component form.

Here, we see this vector 𝐀 drawn on a grid. And we can see the vector starts at the origin of a coordinate frame. Let’s call the horizontal axis the π‘₯-axis and the vertical one the 𝑦. Now, when we go to write this vector 𝐀 in component form, that means we’ll write it in terms of an π‘₯- and a 𝑦-component, also called a horizontal and vertical component. If we call the π‘₯-component of vector 𝐀 𝐴 sub π‘₯ and the 𝑦-component 𝐴 sub 𝑦, then we can multiply each one of these components by the appropriate unit vector. The unit vector for the π‘₯- or horizontal direction is 𝐒 hat, and the unit vector for the vertical or 𝑦-direction is 𝐣 hat. By themselves, the π‘₯- and 𝑦-components of vector 𝐀 are not vectors; they’re scalar quantities. But when we multiply these scalars by a vector, the unit vectors, the result is a vector.

Finally, adding these vector components together, we’ll get the vector 𝐀. Expressing 𝐀 this way is known as writing it in component form. So then, what are 𝐴 sub π‘₯ and 𝐴 sub 𝑦? To figure that out, we’ll need to look at our grid. Starting with 𝐴 sub π‘₯, that’s equal to the horizontal component of this vector 𝐀. In other words, if we project this vector perpendicularly onto the π‘₯-axis, then the length of that line segment, this length here, is 𝐴 sub π‘₯. In terms of the units of this grid, that length is one, two, three units long. And notice that we moved to the left of the origin, that is, into negative π‘₯-values. So even though this horizontal orange line is three units long, we say that the π‘₯-component of 𝐀 is negative three. This is because the projection of vector 𝐀 onto the horizontal axis goes negative three units in the π‘₯-direction.

To find the vertical component of 𝐀, we’ll follow a similar process. Once again, we project vector 𝐀 perpendicularly, this time onto the vertical axis. And it’s the length of this line that tells us the vertical or 𝑦-component of 𝐀. We see that this is one, two units long and that this is in the positive 𝑦-direction. 𝐴 sub 𝑦 then is equal to positive two. And now we can write out 𝐀 in its component form. Vector 𝐀 is equal to negative three times the 𝐒 hat unit vector plus two times the 𝐣 hat unit vector.

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