To evaluate the expression $5\mathbf{a} - 4\mathbf{b} + 3\mathbf{c}$, we will break it down into components:

Given: [ \mathbf{a} = [3, -7, 8], \quad \mathbf{b} = [-6, 3, 4], \quad \mathbf{c} = [2, 5, 7] ]

We first scale each vector by the given coefficients:

[ 5\mathbf{a} = 5 \times [3, -7, 8] = [5 \times 3, 5 \times (-7), 5 \times 8] = [15, -35, 40] ] [ -4\mathbf{b} = -4 \times [-6, 3, 4] = [-4 \times (-6), -4 \times 3, -4 \times 4] = [24, -12, -16] ] [ 3\mathbf{c} = 3 \times [2, 5, 7] = [3 \times 2, 3 \times 5, 3 \times 7] = [6, 15, 21] ]

Now, sum the resulting vectors component-wise:

[ 5\mathbf{a} - 4\mathbf{b} + 3\mathbf{c} = [15, -35, 40] + [24, -12, -16] + [6, 15, 21] ]

Performing the addition:

$x\text{-component}: 15 + 24 + 6 = 45$ $y\text{-component}: -35 + (-12) + 15 = -32$ $z\text{-component}: 40 + (-16) + 21 = 45$

Thus, the resulting vector is: $5\mathbf{a} - 4\mathbf{b} + 3\mathbf{c} = [45, -32, 45]$

Would you like further details on vector operations or any specific clarification?

Here are 5 related questions:

1. How do you find the magnitude of a vector?
2. Can this vector operation be done in terms of dot products?
3. What is the cross product of two vectors?
4. How would you normalize the vector $[45, -32, 45]$?
5. How does vector addition work geometrically?

Tip: When adding vectors, always handle each component separately and perform operations component-wise.