Q. 65 - Calculus

Question

Use the definition of the cross product to prove that the cross product of two parallel vectors is $0$. (This is Theorem 10.26.)

Step-by-Step Solution

Answer

Hence, we prove that $u\times v=0$.

Step 1. Given Information

Use the definition of the cross product to prove that the cross product of two parallel vectors is $0$.

Step 2. Theorem 10.26

Theorem states that, "The cross product of two parallel vectors u and v in ${\mathrm{ℝ}}^3$ is $u\times v=0$."

Step 3. As we know that the cross product of a vector with itself is 0 .

Let $u$ and v be parallel vectors, let there be a scalar $c$ such that $v=cu$. Consider their cross product:

$$\begin{gathered} u\times v=u\times \left(cu\right) \\ u\times v=c\times (u\times u) \\ u\times v=c\times 0 \\ u\times v=0 \end{gathered}$$