Let a, b and c be the vectors [1,2,0], [0, -1, 2] and [2,1,3] respectively.
Find
(a) a· b,
(b) a· (b + c),
(c) 2b· c.

Let's solve the given vector problems step by step.

Given vectors:
\[ \mathbf{a} = [1, 2, 0] \]
\[ \mathbf{b} = [0, -1, 2] \]
\[ \mathbf{c} = [2, 1, 3] \]

### (a) Calculate \(\mathbf{a} \cdot \mathbf{b}\)
The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

For \(\mathbf{a} = [1, 2, 0]\) and \(\mathbf{b} = [0, -1, 2]\):
\[ \mathbf{a} \cdot \mathbf{b} = (1 \cdot 0) + (2 \cdot -1) + (0 \cdot 2) \]
\[ \mathbf{a} \cdot \mathbf{b} = 0 - 2 + 0 \]
\[ \mathbf{a} \cdot \mathbf{b} = -2 \]

### (b) Calculate \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})\)
First, find \(\mathbf{b} + \mathbf{c}\):
\[ \mathbf{b} + \mathbf{c} = [0, -1, 2] + [2, 1, 3] \]
\[ \mathbf{b} + \mathbf{c} = [0+2, -1+1, 2+3] \]
\[ \mathbf{b} + \mathbf{c} = [2, 0, 5] \]

Now, calculate the dot product \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c})\):
\[ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot [2, 0, 5] \]
\[ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = (1 \cdot 2) + (2 \cdot 0) + (0 \cdot 5) \]
\[ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 2 + 0 + 0 \]
\[ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 2 \]

### (c) Calculate \(2\mathbf{b} \cdot \mathbf{c}\)
First, find \(2\mathbf{b}\):
\[ 2\mathbf{b} = 2 \cdot [0, -1, 2] \]
\[ 2\mathbf{b} = [0 \cdot 2, -1 \cdot 2, 2 \cdot 2] \]
\[ 2\mathbf{b} = [0, -2, 4] \]

Now, calculate the dot product \(2\mathbf{b} \cdot \mathbf{c}\):
\[ 2\mathbf{b} \cdot \mathbf{c} = [0, -2, 4] \cdot [2, 1, 3] \]
\[ 2\mathbf{b} \cdot \mathbf{c} = (0 \cdot 2) + (-2 \cdot 1) + (4 \cdot 3) \]
\[ 2\mathbf{b} \cdot \mathbf{c} = 0 - 2 + 12 \]
\[ 2\mathbf{b} \cdot \mathbf{c} = 10 \]

### Summary of Results
(a) \(\mathbf{a} \cdot \mathbf{b} = -2\)

(b) \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = 2\)

(c) \(2\mathbf{b} \cdot \mathbf{c} = 10\)

Would you like any further details or have any questions?

Here are some questions you might be interested in:
1. How do you compute the cross product of two vectors?
2. What is the geometric interpretation of the dot product?
3. How do you find the magnitude of a vector?
4. How can you determine if two vectors are orthogonal?
5. What is the difference between dot product and cross product?

**Tip:** When computing the dot product, make sure to multiply corresponding components and sum them up correctly to avoid mistakes.

Explore the solution to vector problems involving dot products and vector addition. Learn how to compute a·b, a·(b + c), and 2b·c with detailed steps. This guide is perfect for high school students studying vector algebra.

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