Consider three vectors `vecp=hati+hatj+hatk,vecq=2hati+4hatj-hatk and vecr=hati+hatj+3hatk` and let `vecs` be a unit vector, then `vecp,vecq and vecr` are

Options

A. linealy dependent
B. can form the sides of a possible triangle
C. such that the vectors (q-r) is orthogonal to p
D. such that each one of these can be expressed as a linear combination of the other two

Answer

To solve the problem, we need to analyze the three given vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) and determine their relationships, particularly focusing on linear dependence and orthogonality.

### Step-by-Step Solution:

1. **Define the Vectors:**
\[ \vec{p} = \hat{i} + \hat{j} + \hat{k} \]
\[ \vec{q} = 2\hat{i} + 4\hat{j} - \hat{k} \]
\[ \vec{r} = \hat{i} + \hat{j} + 3\hat{k} \]

2. **Calculate \(\vec{q} - \vec{r}\):**
\[ \vec{q} - \vec{r} = (2\hat{i} + 4\hat{j} - \hat{k}) - (\hat{i} + \hat{j} + 3\hat{k}) \]
\[ = (2\hat{i} - \hat{i}) + (4\hat{j} - \hat{j}) + (-\hat{k} - 3\hat{k}) \]
\[ = \hat{i} + 3\hat{j} - 4\hat{k} \]

3. **Check for Orthogonality:**
To check if \(\vec{p}\) is orthogonal to \(\vec{q} - \vec{r}\), we calculate the dot product:
\[ \vec{p} \cdot (\vec{q} - \vec{r}) = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} + 3\hat{j} - 4\hat{k}) \]
\[ = (1)(1) + (1)(3) + (1)(-4) \]
\[ = 1 + 3 - 4 = 0 \]

4. **Conclusion:**
Since the dot product is zero, \(\vec{p}\) is orthogonal to \(\vec{q} - \vec{r}\). This implies that the vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are not linearly independent, and \(\vec{q} - \vec{r}\) is perpendicular to \(\vec{p}\). Therefore, \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) can form the sides of a triangle.

### Final Answer:
The vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are linearly dependent, and \(\vec{q} - \vec{r}\) is orthogonal to \(\vec{p}\).