If 

a⃗ =2i^+j^a→=2i^+j^

, 

b⃗ =2i^−j^+k^b→=2i^−j^+k^

 and 

c⃗ =i^+2k^c→=i^+2k^

. Then 

(c⃗ ×b⃗ ).a⃗ =

We are given the following vectors:
\[
\vec{a} = 2\hat{i} + \hat{j}, \quad \vec{b} = 2\hat{i} - \hat{j} + \hat{k}, \quad \vec{c} = \hat{i} + 2\hat{k}
\]
We are asked to compute \( (\vec{c} \times \vec{b}) \cdot \vec{a} \), which involves two operations:
1. **Cross product**: \( \vec{c} \times \vec{b} \)
2. **Dot product**: \( (\vec{c} \times \vec{b}) \cdot \vec{a} \)

### Step 1: Compute \( \vec{c} \times \vec{b} \)

The cross product \( \vec{c} \times \vec{b} \) is calculated using the determinant of the following matrix:
\[
\vec{c} \times \vec{b} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & 0 & 2 \\
2 & -1 & 1
\end{vmatrix}
\]

Expanding the determinant:
\[
\vec{c} \times \vec{b} = \hat{i} \begin{vmatrix} 0 & 2 \\ -1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 0 \\ 2 & -1 \end{vmatrix}
\]

Calculating each 2x2 determinant:
- For \( \hat{i} \): \( \begin{vmatrix} 0 & 2 \\ -1 & 1 \end{vmatrix} = (0 \times 1) - (2 \times -1) = 2 \)
- For \( \hat{j} \): \( \begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} = (1 \times 1) - (2 \times 2) = 1 - 4 = -3 \)
- For \( \hat{k} \): \( \begin{vmatrix} 1 & 0 \\ 2 & -1 \end{vmatrix} = (1 \times -1) - (0 \times 2) = -1 \)

Thus, the cross product is:
\[
\vec{c} \times \vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}
\]

### Step 2: Compute \( (\vec{c} \times \vec{b}) \cdot \vec{a} \)

Now, take the dot product of \( \vec{c} \times \vec{b} = 2\hat{i} + 3\hat{j} - \hat{k} \) with \( \vec{a} = 2\hat{i} + \hat{j} \):
\[
(\vec{c} \times \vec{b}) \cdot \vec{a} = (2)(2) + (3)(1) + (-1)(0)
\]

Simplifying:
\[
(\vec{c} \times \vec{b}) \cdot \vec{a} = 4 + 3 + 0 = 7
\]

### Final Answer:
\[
(\vec{c} \times \vec{b}) \cdot \vec{a} = 7
\]

Would you like to dive deeper into any specific part of this solution?

### Related Questions:
1. What are the properties of the cross product of two vectors?
2. How does the right-hand rule help in determining the direction of the cross product?
3. What are some physical applications of cross products in physics?
4. How is the dot product used to calculate projections in geometry?
5. Can you interpret the geometric meaning of the scalar triple product \( (\vec{c} \times \vec{b}) \cdot \vec{a} \)?

**Tip:** When dealing with cross products and dot products, keeping track of vector components and signs is crucial for ensuring accuracy.

Learn how to compute (c⃗ × b⃗ ).a⃗ in vector mathematics. Understand the cross product and dot product operations with detailed step-by-step solution. Suitable for advanced high school students and those interested in vector algebra.

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