This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."
Students can define a vector as an arrow of varying length pointed in a particular direction. Student can compose a right triangle from a vector by finding the horizontal and vertical components of the vector. Students can combine vectors in component form or with given magnitudes and directions and finding a resulting magnitude and direction. Student can correct the results of an inverse tangent function based on the quadrant of the vector.
Vectors goes pretty quickly because all the math is a new spin on right triangle stuff. If you spent some time relating cos with horizontal measurements and sin with vertical measurements the breakdown of a vector into components is pretty easy. Some years I've talked about coming up with vector components to describe how to travel from one coordinate point to another. I skipped that this year.
I don't have any particularly interesting intros. It's a lot of drawing arrows with a known magnitude and direction and turning them into right triangles, then just showing them the mag * cos theta or mag * sin theta thing. You could have them draw some arbitrary arrows, do some measuring, and then work back to solving right triangles if you want.
They practice component breakdowns and combining vectors that are already in component form (I use the i, j, and k physics conventions). They figure that out on their own, it's easy.
Show some videos of airplanes landing in crosswinds.
Next we work backwards on finding magnitude and direction given only the components. I spend a moment talking about solving right triangles which will lead them to the use of an inverse tangent. I always discuss angles as relative to zero (getting answers that range from 0-360), and we always work in degree mode. Calculator output needs to be interpreted here, because in quadrant II, III, and IV the calculator is solving a triangle, and the answer will always be less than 90. The answers are also relative to the horizontal, not zero. I emphasize drawing vector systems quite heavily to help with this idea. If the vector is located in quadrant III, but the calculator tells me the direction is 75º, what is that 75º really describing?
Through this discussion we come up with corrections: quadrant I is fine, quad II is ans+180, quad III is ans+180 and quad IV is ans+360. Students will usually fail to correct an angle for a quad III vector because the calculator outputs a positive number. Kids always seem to think positive = success.
Next we do the whole thing in three dimensions. It's limited though. They'll describe the appropriate octant for a 3D vector, and its magnitude. I don't have time to get into the subtleties of angles relative to the three axes. In the future when they're learning with holograms, sure.
Last, we talk about the angles between two given vectors. And no, I don't use the hokey formula with dot products. To start they draw the two vectors and literally measure it. Going further we talk about how to find relative angle measurements. If you're feeling really adventurous you can have them solve the arbitrary triangle created or find its area.
Three activities that crop up. A couple of them are going to change in the future due to changing time constraints.
To stress how a vector system is arranged, I have them create a few in three of the four quadrants. They demonstrate the components and the vector itself using straws cut to scale. An error I've made is rushing this activity before the students have a good idea what I'm talking about. It's also a bit messy and there's nothing necessary about the straws other than giving a physical component.
Since students are bad at visualizing 3D coordinates (they really are, I know you don't believe me), we do some walking around in mathematically defined 3D space. Take some yarn and make an x-y-z axis set in the room. Label the octants. Show the students around and discuss sign conventions. Then have them locate a big set of coordinates printed on labels (set of 60, Word, PDF) inside your 3D coordinate box.
Last, I have them construct their own 3D coordinate system using a bunch of straws and electrical tape. They label each octant and then give me an example of a point that belongs there. Only problem with this one is it is very time consuming (solid 80 minutes start to finish) and might be a little light on learning. I have some ideas about how to improve it next year.
Due to the nature of the components these will fall apart very quickly, some within hours of finishing. Students or are more dutiful with their taping have much better results.
The concepts in vectors span a long period of time even though the material is pretty simple. There are a lot of conceptual things to talk about here. The foundations of this unit come back when discussing polar coordinate systems, so the more time you spend on it the better.
The topics covered span four tests and strike a 50/50 balance between mechanics and concepts.