Question Video: Expressing a Vector in Component Form | Nagwa Question Video: Expressing a Vector in Component Form | Nagwa

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

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Write π in component form.

01:36

### Video Transcript

Write π in component form.

Here, we see this vector π drawn on a grid. And we can see the vector starts at the origin of a coordinate frame. Letβs call the horizontal axis the π₯-axis and the vertical one the π¦. Now, when we go to write this vector π in component form, that means weβll write it in terms of an π₯- and a π¦-component, also called a horizontal and vertical component. If we call the π₯-component of vector π π΄ sub π₯ and the π¦-component π΄ sub π¦, then we can multiply each one of these components by the appropriate unit vector. The unit vector for the π₯- or horizontal direction is π’ hat, and the unit vector for the vertical or π¦-direction is π£ hat. By themselves, the π₯- and π¦-components of vector π are not vectors; theyβ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, weβll get the vector π. Expressing π this way is known as writing it in component form. So then, what are π΄ sub π₯ and π΄ sub π¦? To figure that out, weβll need to look at our grid. Starting with π΄ sub π₯, thatβs equal to the horizontal component of this vector π. In other words, if we project this vector perpendicularly onto the π₯-axis, then the length of that line segment, this length here, is π΄ 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 π is negative three. This is because the projection of vector π onto the horizontal axis goes negative three units in the π₯-direction.

To find the vertical component of π, weβll follow a similar process. Once again, we project vector π perpendicularly, this time onto the vertical axis. And itβs the length of this line that tells us the vertical or π¦-component of π. We see that this is one, two units long and that this is in the positive π¦-direction. π΄ sub π¦ then is equal to positive two. And now we can write out π in its component form. Vector π is equal to negative three times the π’ hat unit vector plus two times the π£ hat unit vector.

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