Find a vector in $\mathbb{R}^4$ that is orthogonal to both $u$ and $v$.

Question

Let $U$ and $V$ be the points in $\mathbb{R}^4$ with position vectors $u=(1, 2, -1, -3)$ and $v=(1, 0, -2, 3)$.

Find a vector in $\mathbb{R}^4$ that is orthogonal to both $u$ and $v$.

I know that $u$ and $v$ are orthogonal to a vector if the dot product of say vector $a=(x,y,z,w)$ and $u$ is $0$ and so is the dot product of vector $a$ and vector $v$.

I get $0=x+2y-z-3w=x-2z+3w$

I don't know how do the rest of this problem for we haven't talked about cross products and thats what I've been seeing online. I would like to ask for some help.

Answer 1

Now you have a system two linear equations in four variables, and you just have to find one solution. What you now can do is plugging in some arbitrary $x$, an arbitrary $z$ (one of them should be nonzero, otherwise $w$ will be zero) and then determine $w$ from the second equation. Finally you can solve for $z$ using the first equation.

Let us use e.g. $x = 1$ and $z=-1$ then the second equation implies $w=-1$. Using the first equation we get $w = -3/2$.

Answer 2

The null space of a matrix is the orthogonal complement of its row space, so compute the null space of the matrix $$\begin{bmatrix}1&2&-1&-3\\1&0&-2&3\end{bmatrix},$$ which has the two given vectors for its rows. Any element of the null space, say one of the basis vectors that you can find via the rref of the matrix, will do.