If `vecA = - 4 hatj + 3 hatj and vecB = 2hati + 5 hatj and vecC = vecA xx vecB`, then the vector `vecC` makes an angle of

Options

A. `45^(@) ` with x - axis
B. `180^(@) ` with y - axis
C. `0^(@) ` with z - axis
D. `180^(@) ` with z - axis

Explanation

To solve the problem, we need to find the vector \(\vec{C}\) which is the cross product of vectors \(\vec{A}\) and \(\vec{B}\), and then determine the angles that \(\vec{C}\) makes with the x, y, and z axes.

### Step 1: Define the vectors
Given:
\[ \vec{A} = -4 \hat{i} + 3 \hat{j} \]
\[ \vec{B} = 2 \hat{i} + 5 \hat{j} \]

### Step 2: Calculate the cross product \(\vec{C} = \vec{A} \times \vec{B}\)
Using the determinant method for the cross product:
\[ \vec{C} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -4 & 3 & 0 \\ 2 & 5 & 0 \end{vmatrix} \]

Calculating the determinant:
\[ \vec{C} = \hat{i} \begin{vmatrix} 3 & 0 \\ 5 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} -4 & 0 \\ 2 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} -4 & 3 \\ 2 & 5 \end{vmatrix} \]

Calculating each of the minors:
1. \(\begin{vmatrix} 3 & 0 \\ 5 & 0 \end{vmatrix} = 3 \cdot 0 - 5 \cdot 0 = 0\)
2. \(\begin{vmatrix} -4 & 0 \\ 2 & 0 \end{vmatrix} = -4 \cdot 0 - 2 \cdot 0 = 0\)
3. \(\begin{vmatrix} -4 & 3 \\ 2 & 5 \end{vmatrix} = (-4)(5) - (3)(2) = -20 - 6 = -26\)

Putting it all together:
\[ \vec{C} = 0 \hat{i} - 0 \hat{j} - 26 \hat{k} = -26 \hat{k} \]

### Step 3: Determine the angles with the axes
1. **Angle with the x-axis**:
\[ \cos \theta_x = \frac{\vec{C} \cdot \hat{i}}{|\vec{C}| |\hat{i}|} = \frac{0}{26 \cdot 1} = 0 \implies \theta_x = 90^\circ \]

2. **Angle with the y-axis**:
\[ \cos \theta_y = \frac{\vec{C} \cdot \hat{j}}{|\vec{C}| |\hat{j}|} = \frac{0}{26 \cdot 1} = 0 \implies \theta_y = 90^\circ \]

3. **Angle with the z-axis**:
\[ \cos \theta_z = \frac{\vec{C} \cdot \hat{k}}{|\vec{C}| |\hat{k}|} = \frac{-26}{26 \cdot 1} = -1 \implies \theta_z = 180^\circ \]

### Final Result
The vector \(\vec{C} = -26 \hat{k}\) makes:
- An angle of \(90^\circ\) with the x-axis,
- An angle of \(90^\circ\) with the y-axis,
- An angle of \(180^\circ\) with the z-axis.