Calculating Cross Product Using Components: Videos & Practice Problems

To calculate the vector product \( \mathbf{A} \times \mathbf{B} \) using unit vector components, first create a table of the x, y, and z components for both vectors. Then, apply the formula \( C_x = A_y B_z - A_z B_y \), \( C_y = A_z B_x - A_x B_z \), and \( C_z = A_x B_y - A_y B_x \) to find each component. This method emphasizes the importance of understanding vector components and their relationships in physics, particularly in contexts like angular momentum and electromagnetic interactions.

Calculating the Vector (Cross) Product Using Components

To calculate the cross product of two vectors, A and B, when given their components in terms of unit vectors (i, j, k), you can follow a systematic approach. The cross product, denoted as A × B, results in a new vector C, which can be expressed in terms of its components: C = C_xi + C_yj + C_zk.

First, identify the components of vectors A and B. For example, if A = i + 2j and B = -2i + 3j + 4k, you can represent them as:

• A_x = 1, A_y = 2, A_z = 0
• B_x = -2, B_y = 3, B_z = 4

Next, create a table to organize these components, repeating the x and y columns for clarity. The next step involves calculating each component of vector C using the formula:

C_x = A_yB_z - B_yA_z

C_y = A_zB_x - B_zA_x

C_z = A_xB_y - B_xA_y

For C_x, substitute the values: C_x = (2)(4) - (3)(0) = 8.

For C_y, substitute: C_y = (0)(-2) - (4)(1) = -4.

For C_z, substitute: C_z = (1)(3) - (-2)(2) = 3 + 4 = 7.

Thus, the resulting vector C can be expressed as:

C = 8i - 4j + 7k.

In summary, the cross product can be calculated using the components of the vectors and following the systematic approach of multiplying the appropriate components diagonally, ensuring to maintain the order of operations. This method provides a clear and consistent way to find the cross product without needing angles or diagrams.

Finding Vector Product Using 2 Methods

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.

Here’s what students ask on this topic:

How do you calculate the cross product of two vectors using their components?

To calculate the cross product of two vectors $A$ and $B$ using their components, follow these steps:

1. Create a table of the x, y, and z components for both vectors.

2. Use the following formulas to find each component of the resulting vector $C$:

$C_x=A_y{B}_z-A_z{B}_y$

$C_y=A_z{B}_x-A_x{B}_z$

$C_z=A_x{B}_y-A_y{B}_x$

3. Combine these components to form the resulting vector $C$ in unit vector notation.

What is the formula for the cross product of two vectors in component form?

The formula for the cross product of two vectors $A$ and $B$ in component form is:

$A\times B=(A_y{B}_z-A_z{B}_y)i+(A_z{B}_x-A_x{B}_z)j+(A_x{B}_y-A_y{B}_x)k$

Why can't you use the formula $ABsin\theta$ to calculate the cross product in terms of components?

The formula $ABsin\theta$ calculates the magnitude of the cross product but does not provide the direction or the components of the resulting vector. When vectors are given in component form (i.e., in terms of $i$, $j$, and $k$), you need to use the component-wise method to find the exact vector result, including its direction and magnitude.

What are the steps to find the cross product of two vectors using the component method?

To find the cross product of two vectors using the component method, follow these steps:

1. List the x, y, and z components of both vectors.

2. Use the formulas:

$C_x=A_y{B}_z-A_z{B}_y$

$C_y=A_z{B}_x-A_x{B}_z$

$C_z=A_x{B}_y-A_y{B}_x$

3. Combine these components to form the resulting vector $C$ in unit vector notation.

What is the significance of the cross product in physics?

The cross product is significant in physics because it is used to determine a vector that is perpendicular to two given vectors. This is crucial in various applications such as calculating torque, angular momentum, and the magnetic force on a moving charge. The direction of the resulting vector follows the right-hand rule, and its magnitude is given by the product of the magnitudes of the original vectors and the sine of the angle between them.