Find $||(x+y) \times (1,1,1)||$

Question

I'm having some trouble with the following question:

Let $x,y \in \mathbb R^3$ such that:

• $||x + y|| = \sqrt{29}$
• $x \times (2,3,4) = (2,3,4) \times y$

Then, find the value of $$||(x+y) \times (1,1,1)||$$

From the information the we are given we know that $$(x + y) \times (2,3,4) = 0$$ meaning that $(x+y) \in span(2,3,4)$, but that us all that I could conclude. How can I solve this?

Answer 1

We have

$$\| (x+y) \times (1,1,1) \| = \|x+y\|\|(1,1,1)\||\sin \alpha|$$

where $\alpha$ is the angle between $(x+y)$ and $(1,1,1)$.

Since you have shown that $(x+y) \in span(2,3,4)$, $\alpha$ is also the angle between $(1,1,1)$ and $(2,3,4)$ (up to modulo $\pi$ or with a minus but we do not care for the norm), thus

$$\|(1,1,1)\times(2,3,4)\| = \|(1,1,1)\|\|(2,3,4)\|| \sin \alpha |$$

Can you conclude ?