Statement 1: If `veca, vecb` are non zero and non collinear vectors, then
`vecaxxvecb=[(veca, vecb, hati)]hati+[(veca, vecb, hatj)]hatj+[(veca, vecb, hatk)]hatk`
Statement 2: For any vector `vecr`
`vecr=(vecr.hati)hati+(vecr.hatj)hatj+(vecr.hatk)hatk`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the given problem, we need to analyze the two statements provided and determine their validity step by step.

### Step 1: Analyze Statement 1
Statement 1 states:
If \(\vec{a}\) and \(\vec{b}\) are non-zero and non-collinear vectors, then:
\[ \vec{a} \times \vec{b} = [(\vec{a}, \vec{b}, \hat{i})] \hat{i} + [(\vec{a}, \vec{b}, \hat{j})] \hat{j} + [(\vec{a}, \vec{b}, \hat{k})] \hat{k} \]

**Explanation:**
- The left-hand side, \(\vec{a} \times \vec{b}\), represents the cross product of vectors \(\vec{a}\) and \(\vec{b}\).
- The right-hand side consists of scalar triple products, which can be expressed as:
- \((\vec{a}, \vec{b}, \hat{i})\) is the scalar triple product of \(\vec{a}\), \(\vec{b}\), and the unit vector in the x-direction \(\hat{i}\).
- Similarly, \((\vec{a}, \vec{b}, \hat{j})\) and \((\vec{a}, \vec{b}, \hat{k})\) represent the scalar triple products with \(\hat{j}\) and \(\hat{k}\) respectively.

**Verification:**
- We can express the cross product \(\vec{a} \times \vec{b}\) in terms of its components:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \]
- This determinant can be expanded to yield components in the \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) directions.
- Thus, the expression on the right-hand side can be shown to equal \(\vec{a} \times \vec{b}\).

**Conclusion:**
- Statement 1 is **True**.

### Step 2: Analyze Statement 2
Statement 2 states:
For any vector \(\vec{r}\):
\[ \vec{r} = (\vec{r} \cdot \hat{i}) \hat{i} + (\vec{r} \cdot \hat{j}) \hat{j} + (\vec{r} \cdot \hat{k}) \hat{k} \]

**Explanation:**
- This statement represents the decomposition of a vector \(\vec{r}\) into its components along the coordinate axes.
- The dot product \(\vec{r} \cdot \hat{i}\) gives the component of \(\vec{r}\) in the direction of \(\hat{i}\), and similarly for \(\hat{j}\) and \(\hat{k}\).

**Verification:**
- The vector \(\vec{r}\) can be expressed in terms of its components as:
\[ \vec{r} = r_1 \hat{i} + r_2 \hat{j} + r_3 \hat{k} \]
- Where \(r_1 = \vec{r} \cdot \hat{i}\), \(r_2 = \vec{r} \cdot \hat{j}\), and \(r_3 = \vec{r} \cdot \hat{k}\).
- Therefore, the equation holds true.

**Conclusion:**
- Statement 2 is **True**.

### Final Conclusion:
Both statements are true, and Statement 2 serves as a correct explanation for Statement 1.

### Final Answer:
Both statements are true, and Statement 2 is a correct explanation for Statement 1.

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