Question

Find all the vectors n= (a,b,c) which are perpendicular to both vectors r1 and r2.
r1=(3,7,8)
r2=(1,8,1)

Answer 1

You can find these vectors by taking the cross product of r1 and r2.

Solution: The cross product of r1 and r2 is
\[r_1\times r_2=\left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\3&7&8\\1&8&1\end{array}\right|=(7-64)\hat{i}-(3-8)\hat{j}+(24-7)\hat{k}=\]
\[-57\hat{i}+5\hat{j}+17\hat{k}=(-57;5;17).\]
So the vectors that are perp. to r1 and r2:
\[\{(-57t;5t;17t;)\mid t\in\mathbb{R}\}.\]