Given the vectors a = [3, -7, 8], b = [-6, 3, 4], and c = [2, 5, 7], evaluate each expression 5a - 4b + 3c

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.

Learn how to evaluate the vector expression 5a - 4b + 3c using scalar multiplication and vector addition. The problem involves vectors a = [3, -7, 8], b = [-6, 3, 4], and c = [2, 5, 7], and results in a new vector [45, -32, 45]. Suitable for students in Grades 9-12.

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