# Plainly by eye, how can 16 year olds visually distinguish $\color{red}{\vec{b} - \vec{r}}$ from $\color{dodgerblue}{|\vec{b}| - |\vec{r}|}$?

Yearly, I teach 16 year olds this diagram beneath (improvement of this) that reappears on standardized tests IN BLACK AND WHITE below with different lengths, letters, and orientation. Tests require students to label in terms of $$\vec{b}, \vec{r}$$ ( = circle's radius ) the:

1. long line from $$(-4, 0)$$ to B.

2. short line segment from a point on the circumference to B. Despite my best endeavors like adding colors, someone ALWAYS mixes up red and blue!

#### How can I better explain their pictoric difference? How would you prevent this diagrammatic mix up BY EYE? • I'm not sure that you can improve the picture much, maybe by adding in an $r'$ which is parallel to $b$ and stating that the norms of $r$ and $r'$ are equal. I'd look to other, similar statements like converting the statement to 1d vectors, i.e. numbers, where the.difference between the two is more elementary.
Oct 1 at 22:40
• Also previously asked here: math.codidact.com/posts/289757
– JRN
Oct 2 at 4:20
• That graphical example is AWFUL! Your point $(4,1)$ is that close to the circle that most students might think it's located ON the circle. Please use points who are less confusing. Oct 2 at 8:33
• @Dominique Are my new pictures better?
– user22729
Oct 3 at 3:42
• This post is very confusing to read, both the OP's text, as well as the fact that the responses (comments and Answers) appear to be referring to previous versions of the Question. What does "this diagram beneath" refer to, is the boldfaced text referring to the B&W diagram or a hypothetical version of it, and why do you call a vector a radius? And why is there a similar-looking question from Codidact; is the OP ('Scexit') user the username 'Tortilla' over there, or are they duplicating someone else's question? Oct 3 at 6:27

I would tell students that absolute value bars represent distance, and distance is indicated by a line segment without an arrow.

This rule of thumb is very intuitive:

• students should already know that absolute value bars remove a negative sign from a number,
• so they should associate bars with the act of "removing things,"
• so it shouldn't feel too weird that bars around a vector remove the arrow,
• and you can make this more rigorous by explaining that in both cases, the bars are just removing the direction from the object that they are enclosing.

Additionally, this rule of thumb provides sufficient information for students to distinguish between $$\mathbf{b}-\mathbf{r}$$ vs $$|\mathbf{b}|-|\mathbf{r}|$$ in the diagram: there is only one expression with absolute value bars, and only one line segment without an arrow, so they must pair up.

• Thanks, but I edited my post. I forgot to include the test's diagram, to show that the question is harder. The test's diagram never draws arrows on the 2 lines in question, which would make this question too straightforward!
– user22729
Oct 1 at 23:25

No one diagram is going to help every student actually absorb this idea. In fact, the mistake reminds me a bit of students thinking $$\sqrt{a^{2}+b^{2}}=a+b$$. They are "simplifying" in a way that makes no real sense, partly because they are not trying to make sense when they're doing math, sadly. Many students think of math as a bunch of rules, and that doesn't really work.

I'd guess they need to work more with each part of this separately: meanings of vectors, subtraction of vectors, and lengths of vectors.

My first reaction is to doubt the problem is that they somehow can't see the difference (how could you miss the vertical lines?), but rather that labeling only one vector with its length is throwing them off. I know it doesn't have an arrow attached but still, it's a little confusing. For that matter, the arrow placements themselves, at least for $$b$$, might cause confusion. What if you just wrote "the length of this segment is $$|b|-|r|$$"?

I guess another explanation could be that their understanding of the math is shaky, in which case they'll probably make all sorts of errors no matter how good the diagrams are.

• I don't think I'd change my answer much. Does the question really ask them to label the line from $(4,1)$ to $B$, or does it ask for the length of the segment? The latter phrasing is much clearer. If I had to redraw the diagram with the length labeled, I would put curly braces under the segment. Oct 1 at 23:52