In this problem, we explore the calculation of the cross product of two vectors, a and b, using two different methods: the sine of the angle between them and the unit vector component approach. The vectors are defined as a = 4i + 3j and b = -2i + 3j.

To find the magnitude of the cross product, we start with the formula for the magnitude of the cross product, given by:

|c| = |a| |b| \sin(\theta)

Here, |c| represents the magnitude of the cross product, |a| and |b| are the magnitudes of vectors a and b, and \theta is the angle between them.

First, we calculate the magnitudes of the vectors:

For vector a:

|a| = \sqrt{(4^2 + 3^2)} = \sqrt{16 + 9} = 5

For vector b:

|b| = \sqrt{((-2)^2 + 3^2)} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6

Next, we need to determine the angle \theta between the two vectors. We can find the angles \theta_a and \theta_b using the inverse tangent function:

\theta_a = \tan^{-1}\left(\frac{3}{4}\right) \approx 36.9^\circ

\theta_b = \tan^{-1}\left(\frac{3}{-2}\right) \approx 56.3^\circ

Thus, the angle between the vectors is:

\theta = 180^\circ - \theta_a - \theta_b \approx 180^\circ - 36.9^\circ - 56.3^\circ \approx 86.8^\circ

Substituting the values into the magnitude formula gives:

|c| = 5 \times 3.6 \times \sin(86.8^\circ \approx 18

To determine the direction of the cross product, we apply the right-hand rule. By pointing the fingers of our right hand along vector a and curling them towards vector b, our thumb points out of the page, indicating that the direction of vector c is along the positive z axis, or k hat.

In the second method, we express the cross product using the unit vector components. We set up a table to extract the components of vectors a and b:

a = (4, 3, 0) and b = (-2, 3, 0).

Using the determinant method for the cross product, we calculate:

c_x = a_y b_z - a_z b_y = 3 \cdot 0 - 0 \cdot 3 = 0

c_y = a_z b_x - a_x b_z = 0 \cdot (-2) - 4 \cdot 0 = 0

c_z = a_x b_y - a_y b_x = 4 \cdot 3 - 3 \cdot (-2) = 12 + 6 = 18

Thus, the cross product vector is c = 0i + 0j + 18k, confirming that the magnitude is 18 and the direction is along the positive z axis.

In conclusion, both methods yield the same result: the magnitude of the cross product is 18, and it points in the k hat direction, confirming the consistency of the calculations.