## Thomas' Calculus 13th Edition

No two are perpendicular. ${\bf u}$ and ${\bf w}$ are parallel.
Two vectors are perpendicular if their dot product is zero. Two vectors are parallel if their cross product is the zero vector. ${\bf u}$ = $\langle 5,-1,1\rangle,\quad {\bf v}$= $\langle 0,1,-5\rangle,\quad {\bf w}$= $\langle-15,3,-3\rangle$ ${\bf u}\cdot{\bf v}=5(0)+(-1)(1)+(1)(-5)=-6\neq 0$ ${\bf u}\cdot{\bf w}=5(-15)+(-1)(3)+(1)(-3)=-81\neq 0$ ${\bf v}\cdot{\bf w}=0(-15)+(1)(3)+(-5)(-3)=-11\neq 0$ No two are perpendicular. ${\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 5 & -1 & 1\\ 0 & 1 & -5 \end{array}\right|=(5-1){\bf i}....\neq{\bf 0}$ ${\bf u}\times{\bf w}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 5 & -1 & 1\\ -15 & 3 & -3 \end{array}\right|=(3-3){\bf i}-(-15+15){\bf j}+(15-15){\bf k}= {\bf 0}$ ${\bf v}\times{\bf w}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 0 & 1 & 5\\ -15 & 3 & -3 \end{array}\right|=(-3-15){\bf i}....\neq{\bf 0}$ ${\bf u}$ and ${\bf w}$ are parallel.