Q. 19 - Calculus

Question

If u and v are vectors in ${\mathrm{ℝ}}^3$ such that $u·v=0$ and $u\times v=0$, what can we conclude about u and v?

Step-by-Step Solution

Answer

$u·v=0 \mathrm{and} u\times v=0$ concluded that at least one of them is $0$.

Step 1. Given Information

If u and v are vectors in ${\mathrm{ℝ}}^3$ such that $u·v=0$ and $u\times v=0$, what can we conclude about u and v?

Step 2. If u and v are vectors in ℝ 3 such that u · v = 0   and   u × v = 0 .

$u·v=0$ if vectors u and v are orthogonal.

The cross product $u\times v=0$ is u and v are parallel.

Now we can also say that $u·v=0 \mathrm{and} u\times v=0$concluded that at least one of them is $0$.