Math Snap

STEP 1

1. We are working in a vector space over the field $\mathbb{R}$.
2. Vector addition and scalar multiplication are defined with specific operations:
- Vector addition: $u \boxplus v := u + v + 2$
- Scalar multiplication: $a \boxdot u := au + 2a - 2$
3. We need to find the sum, scalar multiple, zero vector, and additive inverse using these operations.

STEP 2

1. Calculate the sum using the defined vector addition.
2. Calculate the scalar multiple using the defined scalar multiplication.
3. Determine the zero vector for the vector space.
4. Find the additive inverse of a vector.

STEP 3

Calculate the sum $6 \boxplus (-5)$ using the defined vector addition:
$$6 \boxplus (-5) = 6 + (-5) + 2$$ Perform the arithmetic:
$$6 + (-5) + 2 = 1 + 2 = 3$$ The sum is:
$$\boxed{3}$$

STEP 4

Calculate the scalar multiple $7 \boxdot 6$ using the defined scalar multiplication:
$$7 \boxdot 6 = 7 \cdot 6 + 2 \cdot 7 - 2$$ Perform the arithmetic:
$$42 + 14 - 2 = 54$$ The scalar multiple is:
$$\boxed{54}$$

STEP 5

Determine the zero vector $\underline{0}_{V}$ such that for any vector $u$, $u \boxplus \underline{0}_{V} = u$.
Using the vector addition definition:
$$u \boxplus \underline{0}_{V} = u + \underline{0}_{V} + 2 = u$$ To satisfy this, $\underline{0}_{V} = -2$.
The zero vector is:
$$\boxed{-2}$$

Find the additive inverse $\boxminus x$ such that $x \boxplus (\boxminus x) = \underline{0}_{V}$.
Using the vector addition definition:
$$x \boxplus (\boxminus x) = x + (\boxminus x) + 2 = -2$$ Solving for $\boxminus x$:
$$x + (\boxminus x) + 2 = -2$$ $$\boxminus x = -2 - 2 - x$$ $$\boxminus x = -x - 4$$ The additive inverse is:
$$\boxed{-x - 4}$$