The image contains the following problem:

In the equation $\vec{F} = q\vec{v} \times \vec{B}$, given that $q = 4$, $\vec{v} = 4\hat{i} + 6\hat{j} + 8\hat{k}$, and $\vec{F} = 4\hat{i} - 20\hat{j} + 24\hat{k}$, calculate the value of $\vec{B}$ in unit vector notation if $B_x = B_y$.

Steps to solve the problem:

1. The equation $\vec{F} = q\vec{v} \times \vec{B}$ is the vector cross product form of the Lorentz force. First, express the force equation: $\vec{F} = q (\vec{v} \times \vec{B})$ Substituting $q = 4$: $\vec{F} = 4 (\vec{v} \times \vec{B})$

2. Now, divide both sides of the equation by 4: $\frac{\vec{F}}{4} = \vec{v} \times \vec{B}$

3. The vectors involved are: $\vec{v} = 4\hat{i} + 6\hat{j} + 8\hat{k}, \quad \vec{F} = 4\hat{i} - 20\hat{j} + 24\hat{k}$

   Let $\vec{B} = B_x\hat{i} + B_x\hat{j} + B_z\hat{k}$ (since $B_x = B_y$).

4. Perform the cross product $\vec{v} \times \vec{B}$: $\vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 6 & 8 \\ B_x & B_x & B_z \end{vmatrix}$

5. Calculate the determinant: [ \vec{v} \times \vec{B} = \hat{i}[(6)(B_z) - (8)(B_x)] - \hat{j}[(4)(B_z) - (8)(B_x)] + \hat{k}[(4)(B_x) - (6)(B_x)] ] Simplifying: $\vec{v} \times \vec{B} = \hat{i}(6B_z - 8B_x) - \hat{j}(4B_z - 8B_x) + \hat{k}(4B_x - 6B_x)$ $\vec{v} \times \vec{B} = \hat{i}(6B_z - 8B_x) - \hat{j}(4B_z - 8B_x) + \hat{k}(-2B_x)$

6. Now, equate $\vec{v} \times \vec{B}$ to $\frac{1}{4}\vec{F} = \hat{i} + -5\hat{j} + 6\hat{k}$: $(6B_z - 8B_x)\hat{i} = 1\hat{i}$ $-(4B_z - 8B_x)\hat{j} = -5\hat{j}$ $-2B_x\hat{k} = 6\hat{k}$

7. Solve these three equations:

   • From $-2B_x = 6$, solve for $B_x$: $B_x = -3$
   • Substituting $B_x = -3$ into the first equation: $6B_z - 8(-3) = 1 \implies 6B_z + 24 = 1 \implies 6B_z = -23 \implies B_z = -\frac{23}{6}$
   • Substituting $B_x = -3$ into the second equation: $4B_z - 8(-3) = 5 \implies 4B_z + 24 = 5 \implies 4B_z = -19 \implies B_z = -\frac{19}{4}$

Thus, the magnetic field vector $\vec{B}$ in unit vector notation is: $\vec{B} = -3\hat{i} - 3\hat{j} - \frac{19}{4}\hat{k}$

Would you like to explore any specific part of this solution further or have any questions?

Here are some related questions:

1. How does the cross product formula for vectors work geometrically?
2. What is the physical significance of the Lorentz force equation?
3. What are the conditions for using the right-hand rule in vector cross products?
4. How would this problem change if $B_x \neq B_y$?
5. What are the units of $\vec{F}$, $q$, $\vec{v}$, and $\vec{B}$ in the SI system?

Tip:

Always check the units in vector calculations to ensure consistency, especially in physics problems.