Show that the vectors :
`vec(a)=hat(i)-2hat(j)+3hat(k), vec(b)=-2hat(i)+3hat(j)-4hat(k)` and `vec(c )=hat(i)-3hat(j)+5hat(k)` are coplanar.

Answer

To show that the vectors \(\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}\), \(\vec{b} = -2\hat{i} + 3\hat{j} - 4\hat{k}\), and \(\vec{c} = \hat{i} - 3\hat{j} + 5\hat{k}\) are coplanar, we can use the scalar triple product. The vectors are coplanar if the scalar triple product \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\).

### Step 1: Calculate the cross product \(\vec{b} \times \vec{c}\)

The cross product of two vectors \(\vec{b}\) and \(\vec{c}\) can be calculated using the determinant of a matrix formed by the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) and the components of \(\vec{b}\) and \(\vec{c}\):

\[ \vec{b} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -2 & 3 & -4 \\ 1 & -3 & 5 \end{vmatrix} \]

### Step 2: Calculate the determinant

Using the determinant formula, we can expand it as follows:

\[ \vec{b} \times \vec{c} = \hat{i} \begin{vmatrix} 3 & -4 \\ -3 & 5 \end{vmatrix} - \hat{j} \begin{vmatrix} -2 & -4 \\ 1 & 5 \end{vmatrix} + \hat{k} \begin{vmatrix} -2 & 3 \\ 1 & -3 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. For \(\hat{i}\):
\[ \begin{vmatrix} 3 & -4 \\ -3 & 5 \end{vmatrix} = (3)(5) - (-4)(-3) = 15 - 12 = 3 \]

2. For \(\hat{j}\):
\[ \begin{vmatrix} -2 & -4 \\ 1 & 5 \end{vmatrix} = (-2)(5) - (-4)(1) = -10 + 4 = -6 \]

3. For \(\hat{k}\):
\[ \begin{vmatrix} -2 & 3 \\ 1 & -3 \end{vmatrix} = (-2)(-3) - (3)(1) = 6 - 3 = 3 \]

Putting it all together:
\[ \vec{b} \times \vec{c} = 3\hat{i} + 6\hat{j} + 3\hat{k} \]

### Step 3: Calculate the dot product \(\vec{a} \cdot (\vec{b} \times \vec{c})\)

Now we will calculate the dot product of \(\vec{a}\) and \(\vec{b} \times \vec{c}\):

\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = (1)(3) + (-2)(6) + (3)(3) \]

Calculating this gives:
\[ = 3 - 12 + 9 = 0 \]

### Conclusion

Since \(\vec{a} \cdot (\vec{b} \times \vec{c}) = 0\), the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar.

---