Find a unit vector in the plane of the vectors $A = i + 2j$ and $B = j + 2k$, perpendicular to the vector $C = 2i + j +2k$.
I'm confused as to what the problem is telling me.
I believe this problem is telling me that $A$ and $B$ span a particular plane and C is normal to that plane. As such $A \times B$ should give me the normal to the plane but $A \times B = 4i – 2j + k \neq 2i +j + 2k = C$.
So how is it possible for these particular $A$ and $B$ to span this plane when this particular $C$ is not normal, even though it is supposed to be?
Best Answer
We note that the vectors $$\vec{a}=(1,2,0) \mbox{ and } \vec{b}=(0,1,2)$$
are linearly independent, and thus span a plane. If you are unfamiliar with these words, I hyperlinked the (possibly) problematic phrases to their respective Wikipedia articles.
Therefore, if we want to find a vector in the plane spanned by $\vec{a}$ and $\vec{b}$, say $\vec{d}$, then $$\vec{d} = \lambda \vec{a} + \mu \vec{b}, \mbox{ for some } \lambda,\mu\in\mathbb{R}.$$
Furthermore, we want $\vec{d}$ to be perpendicular to $\vec{c}=(2,1,2)$. That is, $\vec{d}\cdot \vec{c} = 0. $ So, we see that $$(\lambda \vec{a} + \mu \vec{b})\cdot \vec{c}=\lambda(\vec{a}\cdot\vec{c})+\mu(\vec{b}\cdot\vec{c})=4\lambda + 5\mu=0.$$
We then simply choose $\lambda$ and $\mu$ such that this is the case. For example, $\lambda=5$ and $\mu=-4$. So, $$\vec{d} = 5 \vec{a} -4 \vec{b} = (5,6,-8).$$ Now, we need to normalize the vector; the length of $\vec{d}$ is $\sqrt{25 + 36 + 64}=5\sqrt{5}$. Therefore, the vector $$\frac{1}{5\sqrt{5}} (5,6,-8)$$
satisfies the requirements.