Answer
$\langle -3a, 3b, c \rangle$, $\langle 5a, b, 5c \rangle$, $\langle 5a, \frac{11b}{2}, 10c \rangle$
Work Step by Step
Step 1. Identify the given vectors: $\vec u=\langle a, 2b, 3c \rangle$ and $\vec v=\langle -4a, b, -2c \rangle$
Step 2. Use the algebraic formula for vectors, we have $\vec u+\vec v=\langle a-4a, 2b+b, 3c-2c \rangle
=\langle -3a, 3b, c \rangle$
Step 3. Similarly $\vec u-\vec v=\langle a+4a, 2b-b, 3c+2c \rangle=\langle 5a, b, 5c \rangle$
Step 4. $3\vec u-\frac{1}{2}\vec v=\langle 3a+2a, 6b-\frac{b}{2}, 9c+c \rangle=\langle 5a, \frac{11b}{2}, 10c \rangle$
