Geometric Relationship between Two Vectors

Question

Consider two column vectors such that $a = (1,2,3)^T$ and $b = (-3,3,-1)^T$.

What is the geometric relationship between $a$ and $b$?

Answer 1

Because the dot product vanishes, the vectors are perpendicular.

Answer 2

The dot product of two vectors is $$\left(\begin{matrix}a_1\\a_2\\a_3\end{matrix}\right)\cdot\left(\begin{matrix}b_1\\b_2\\b_3\end{matrix}\right)=a_1b_1+a_2b_2+a_3b_3=|a||b|\cos\theta.$$ Where $|a|$ is the length of the vector $a$.

Try computing $a_1b_1+a_2b_2+a_3b_3$ and see what your result is. Using $|a||b|\cos\theta$, can you interpret your result?