Choosing the vector with length 1

While there are several vectors possible on the Z-axis to choose from, what is the reasoning behind choosing the one with magnitude of 1?

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I am not familiar with this part of the course but a magnitude of 1 is always simpler to understand operations being performed in it.

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The vector with magnitude of 1 is also known as the unit vector, and is usually chosen to focus on the direction rather than magnitude or scale. As @gent.spah mentioned, it can also make it simpler to understand operations, like dot products and projections, being performed on it.

In this context, unit vectors can be used to calculate the distance of a point projected along an axis. In the example given in the screenshot you shared, taking the dot product between the point on the x_1, x_2 axes and the z-axis unit vector gives the distance of that point when projected to the z-axis.

I hope this helps!

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Hello, @abhilash341,

To further elaborate the point of @gent.spah and @Joseph_Kim1 on projection, let’s think about calculating the projection length of a human on an inclined wall

Next to my very ugly drawing, in that simpler diagram, we can easily tell that length should be hcos\theta.

What if it put the diagram onto a vector system, replacing the human with \vec{h} and the wall with the unit vector \hat{z}?

The math tells us if we use the unit vector and not just any arbitrary length of vector along the z direction, we can recover the same projection length with dot product. This is why, when we want projection length on an axis, we do dot product with the unit vector pointing along the axis.

Cheers,
Raymond

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Thank you for your response, Raymond. The primary advantage of using a unit vector is that it simplifies the computations for projecting onto the Z-axis. However, it does not actually change the values recovered for the original axes based on this projection, correct?

Hey, @abhilash341, you are welcome! I am just not quite understanding what you mean for the following part of your question:

maybe you can draw something?