# How is the vector $v-w$ the third side of a right triangle?

I am self studying linear algebra to get ahead before I take it in college, and I found something in the textbook that I'm using and on Khan Academy that has confused me and brought my self learning to a screeching halt.

The following image has been taken from Gilbert Strang's Introduction to Linear Algebra

Prior to this image it saids that when vectors $v$ and $w$ are perpendicular, they form two sides of a right triangle. The third side is $v-w$ (the hypotenuse going across in Figure 1.8) Which is the image I posted. I don't understand how $v-w$ is the hypotenuse. Intuitively, it makes no sense.

In addition, I also watched the Khan Academy Video on this topic, and found that around $2:00$ into the video Sal draws a similar picture and I have no idea how the vector $a-b$ was positioned where it was.

What is the reasoning behind Sal's and Professor Strang's diagrams?

• $v-w$ is the vector that points from $w$ to $v$ because if you start from $w$ and add the vector $v-w$ you end up at $v$ (i.e. $w+(v-w)=v$). Think about it as what you need to add to $w$ to get to $v$. – TravisJ Sep 25 '15 at 22:39
• Are you comfortable with finding resultant vectors geometrically? Notice that $\vec{v}$ is the resultant of adding $\vec{w}$ and the unnamed vector, which I'll call $\vec{x}$. Then $\vec{w}+\vec{x}=\vec{v}$. Solving for $\vec{x}$ gets you $\vec{v}-\vec{w}$. – Tim Thayer Sep 25 '15 at 22:40
• @TimThayer Wow. I can't believe I couldn't figure something that simple out. Your explanation was very concise. If you will write it as the answer to my question, I'll check you off and +1. – Cherry_Developer Sep 25 '15 at 22:44

In the diagram, notice that $\vec{v}$ is the resultant of adding $\vec{w}$ and the unnamed vector, which I'll call $\vec{x}$.
Then $\vec{w}+\vec{x}=\vec{v}$.
Solving for $\vec{x}$ gets you $\vec{v}-\vec{w}$.
The quick way to remember this is the idea of "terminal $-$ initial", which is used to write a vector between two points. Looking at the diagram, $\vec{x}$ terminates at $\vec{v}$ and starts at $\vec{w}$. Thus $\vec{v}-\vec{w}$.