Find all vectors that are perpendicular to both vectors $a = i + 2j + 3k$ and $b = -2i + 2j - 4k$.

1. Problem Description

Find all vectors that are perpendicular to both vectors $a = i + 2j + 3k$ and $b = -2i + 2j - 4k$.

2. Solution Steps

A vector that is perpendicular to two given vectors is proportional to the cross product of those two vectors. Let $v$ be a vector perpendicular to both $a$ and $b$. Then $v = c(a \times b)$ for some scalar $c$.

First, compute the cross product $a \times b$:

$a \times b = \begin{vmatrix} i & j & k \\ 1 & 2 & 3 \\ -2 & 2 & -4 \end{vmatrix}$

$a \times b = i(2(-4) - 3(2)) - j(1(-4) - 3(-2)) + k(1(2) - 2(-2))$

$a \times b = i(-8 - 6) - j(-4 + 6) + k(2 + 4)$

$a \times b = -14i - 2j + 6k$

Therefore, any vector perpendicular to both $a$ and $b$ is of the form $c(-14i - 2j + 6k)$, where $c$ is a scalar. We can also write this as $-2c(7i + j - 3k)$.

3. Final Answer

The vectors perpendicular to both $a$ and $b$ are given by $c(-14i - 2j + 6k)$ or $-2c(7i + j - 3k)$ for any scalar $c$. Equivalently, the set of vectors perpendicular to both $a$ and $b$ is given by $\{c(-14, -2, 6) | c \in \mathbb{R} \}$, or $\{c(7, 1, -3) | c \in \mathbb{R} \}$.