# Finding the value of the cross product of 2 vectors without knowing the value of the vectors?

Let this be the question:

Suppose $$\vec{v}$$ and $$\vec{w}$$ are two vectors parallel to the plane $$x + 2y + 3z = 7.$$ Suppose furthermore that $$\vec{v}$$ is perpendicular to $$\vec{w}$$, $$‖v‖= 3, \ ‖w‖= 4.$$

How would you go about answering this question? I always reach a dead end when I try to solve $$\vec{v}$$ and $$\vec{w}$$ that satisfy the given information, I just don't know how to go about it. I tried to draw the plane and two parallel vectors, then I know the length of the cross product would be $$12$$. Then what do I do now? How could you solve this to find vectors $$\vec{v}$$ and $$\vec{w}$$?

• The cross product is perpendicular to the plane. Therefore it is parallel to the vector (1,2,3) which is perpendicular to the plane. Commented Feb 26, 2022 at 4:18
• @kmitov I see, so the cross product of ⟨v⟩ and ⟨w⟩ would be n(1,2,3), where n is some arbitrary value. we know the length is 12. What do you think I should do next? equate the equation with something?
– mb3
Commented Feb 26, 2022 at 4:28

You have: $$\vec{u}\times \vec{v}= n(1,2,3)\implies |\vec{u}\times\vec{v}|=|n|\sqrt{1^2+2^2+3^2}=|n|\sqrt{14}=12\implies n = \pm\dfrac{12}{\sqrt{14}}\implies \vec{u}\times\vec{v}=\pm\dfrac{12}{\sqrt{14}}\left(1,2,3\right)$$.