Math Snap

STEP 1

1. We are given vectors $\bar{u}$, $\bar{v}$, and $\bar{w}$.
2. The notation $<\cdot,\cdot>$ represents the dot product of two vectors.
3. We need to find the dot product $<3 \bar{u}, 4 \bar{v} + 5 \bar{w}>$.

STEP 2

1. Distribute the scalar multiplication inside the dot product.
2. Apply the distributive property of the dot product.
3. Use the given dot product values to compute the result.

STEP 3

Start by distributing the scalar multiplication inside the dot product:
$$<3 \bar{u}, 4 \bar{v} + 5 \bar{w}> = 3 <\bar{u}, 4 \bar{v} + 5 \bar{w}>$$

STEP 4

Apply the distributive property of the dot product:
$$3 <\bar{u}, 4 \bar{v} + 5 \bar{w}> = 3 (4 <\bar{u}, \bar{v}> + 5 <\bar{u}, \bar{w}>)$$

STEP 5

Substitute the given values of the dot products into the equation:
$$3 (4 \cdot 6 + 5 \cdot (-1))$$

STEP 6

Calculate the expression inside the parentheses:
$$3 (24 - 5) = 3 \times 19$$

Compute the final result:
$$3 \times 19 = 57$$ The solution is:
$$\boxed{57}$$