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The vector cross product is a fundamental operation used in physics and engineering to find a vector that is perpendicular to two other vectors. In the context of magnetic forces, it allows us to determine the direction and magnitude of the force experienced by a charged particle in a magnetic field. The calculation involves taking the cross product of two vectors: the velocity of the charged particle (\(\vec{v}\)) and the magnetic field (\(\vec{B}\)). The expression for the magnetic force is given by:

• \(\vec{F} = q (\vec{v} \times \vec{B})\)

where \(q\) is the charge of the particle.

The cross product of two vectors \(\vec{A} = a_1\, \hat{\mathrm{i}} + a_2\, \hat{\mathrm{j}} + a_3\, \hat{\mathrm{k}}\) and \(\vec{B} = b_1\, \hat{\mathrm{i}} + b_2\, \hat{\mathrm{j}} + b_3\, \hat{\mathrm{k}}\) is calculated using the determinant of a matrix:

• \[\vec{A} \times \vec{B} = \begin{vmatrix} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]

This breakdown leads to a new vector that contains components in the \(\hat{\mathrm{i}}\), \(\hat{\mathrm{j}}\), and \(\hat{\mathrm{k}}\) directions, found by solving the determinant.

In physics, vectors are quantities that have both magnitude and direction. Vector notation is a way of representing these quantities mathematically using unit vectors. A unit vector is a vector of length one that points in a specific direction. Common unit vectors include:

• \(\hat{\mathrm{i}}\) for the x-direction
• \(\hat{\mathrm{j}}\) for the y-direction
• \(\hat{\mathrm{k}}\) for the z-direction

The vector used in magnetic force calculations, such as velocity (\(\vec{v}\)) or magnetic field (\(\vec{B}\)), is expressed as a combination of these unit vectors along with the respective scalar components. For instance, the vector
\(\vec{v} = 2.0 \hat{\mathrm{i}} + 4.0 \hat{\mathrm{j}} + 6.0 \hat{\mathrm{k}}\)indicates that it has components of 2, 4, and 6 in the x, y, and z directions, respectively. Using vector notation simplifies the manipulation of vectors and makes the mathematics more understandable and manageable to work with during problem-solving.

Solving physics problems, like the one dealing with vectors in a magnetic field, involves a combination of understanding the physics principles at play and performing accurate mathematical calculations. The steps are generally:

• Identify known quantities and what you need to determine. Establish the connections between them using relevant formulas. Here, the formula \(\vec{F} = q(\vec{v} \times \vec{B})\) helps us find \(\vec{B}\).
• Set up the mathematical operation needed, such as the cross product in this exercise. Convert the problem into mathematical equations using vector notation.
• Perform the calculations with care, adjusting for any specific conditions stated, like setting \(B_x = B_y\) here.
• Re-examine your solution, especially if there are physical constraints or conditions, ensuring the results make sense in the context of the problem.

By systematically using these steps, you can solve complex physics problems by breaking them into more manageable parts, aiding understanding and ensuring accurate solutions.