Statement 1: Let `vecr` be any vector in space. Then, `vecr=(vecr.hati)hati+(vecr.hatj)hatj+(vecr.hatk)hatk`
Statement 2: If `veca, vecb, vecc` are three non-coplanar vectors and `vecr` is any vector in space then
`vecr={([(vecr, vecb, vecc)])/([(veca, vecb, vecc)])}veca+{([(vecr, vecc, veca)])/([(veca, vecb, vecc)])}vecb+{([(vecr, veca, vecb)])/([(veca, vecb, vecc)])}vecc`

Options

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

Explanation

To solve the problem, we need to analyze both statements and verify their correctness step by step.

### Step 1: Understanding Statement 1
Statement 1 claims that any vector \( \vec{r} \) in space can be expressed as:
\[ \vec{r} = (\vec{r} \cdot \hat{i}) \hat{i} + (\vec{r} \cdot \hat{j}) \hat{j} + (\vec{r} \cdot \hat{k}) \hat{k} \]
This expression represents the projection of the vector \( \vec{r} \) onto the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) along the x, y, and z axes respectively. This is a standard representation of a vector in three-dimensional space.

### Step 2: Understanding Statement 2
Statement 2 states that if \( \vec{a}, \vec{b}, \vec{c} \) are three non-coplanar vectors and \( \vec{r} \) is any vector in space, then:
\[ \vec{r} = \frac{[\vec{r}, \vec{b}, \vec{c}]}{[\vec{a}, \vec{b}, \vec{c}]} \vec{a} + \frac{[\vec{r}, \vec{c}, \vec{a}]}{[\vec{a}, \vec{b}, \vec{c}]} \vec{b} + \frac{[\vec{r}, \vec{a}, \vec{b}]}{[\vec{a}, \vec{b}, \vec{c}]} \vec{c} \]
Here, \( [\vec{x}, \vec{y}, \vec{z}] \) denotes the scalar triple product of the vectors \( \vec{x}, \vec{y}, \vec{z} \). This statement is a representation of the vector \( \vec{r} \) in terms of the non-coplanar vectors \( \vec{a}, \vec{b}, \vec{c} \).

### Step 3: Proving Statement 2
To prove Statement 2, we can express \( \vec{r} \) as a linear combination of the non-coplanar vectors:
\[ \vec{r} = x \vec{a} + y \vec{b} + z \vec{c} \]
where \( x, y, z \) are scalars. By taking the cross products of \( \vec{r} \) with the other vectors, we can derive the coefficients \( x, y, z \) in terms of the scalar triple products.

1. **Cross Product with \( \vec{b} \) and \( \vec{c} \)**:
\[ \vec{r} \times (\vec{b} \times \vec{c}) = x \vec{a} \times (\vec{b} \times \vec{c}) + y \vec{b} \times (\vec{b} \times \vec{c}) + z \vec{c} \times (\vec{b} \times \vec{c}) \]
Using the properties of cross products, we can isolate \( x \) as:
\[ x = \frac{[\vec{r}, \vec{b}, \vec{c}]}{[\vec{a}, \vec{b}, \vec{c}]} \]

2. **Cross Product with \( \vec{c} \) and \( \vec{a} \)**:
Similarly, we can find \( y \):
\[ y = \frac{[\vec{r}, \vec{c}, \vec{a}]}{[\vec{a}, \vec{b}, \vec{c}]} \]

3. **Cross Product with \( \vec{a} \) and \( \vec{b} \)**:
Finally, we can find \( z \):
\[ z = \frac{[\vec{r}, \vec{a}, \vec{b}]}{[\vec{a}, \vec{b}, \vec{c}]} \]

Thus, we have shown that Statement 2 is indeed a correct representation of \( \vec{r} \) in terms of the non-coplanar vectors \( \vec{a}, \vec{b}, \vec{c} \).

### Conclusion
Both statements are true. Statement 2 provides a correct explanation of Statement 1.