Triangle properties from vectors

Question

I am a beginner at linear algebra and I am still trying to connect the dots. I stumbled upon this problem and I had no idea how can I approach it. I would appreciate any help. Thanks!

Answer 1

There are 2 ways to approach linear algebra: Visually and algebraically. If you're struggling I suggest to try the visual approach. In this case: Try to simplify the problem!

A triangle is 2 dimensional, so it exists on a plane, that's enough to give you a decent visualisation. Do you know what the length of each of these vectors is? How would that affect the triangle?

Answer 2

HINT: For a point A$$A={ \begin{bmatrix}A_x\\A_y\\A_z \end{bmatrix}}$$ Giving You $$\vec A=-2\vec i-6\vec j+3\vec k$$ $$\vec B=\vec i+3\vec j-12\vec k$$ $$\vec C=\vec i+3\vec j+9\vec k$$

Hope it explains enough about your triangle?

Answer 3

You can figure out the internal angles of the triangles for one $A\cdot B = |A||B|\cos\theta$ (Which will tell you if it's equilateral/isosceles etc). You can also figure out its area $\frac{1}{2}|\vec{AB}\times\vec{AC}|$ etc. I'm not entirely sure what the question meant as it was pretty vague, but those are the first things that come to mind.