Answer:
The value of k is -45, and the magnitude of the magnetic force F is 5 * sqrt(2650).
Step-by-step explanation:
To find k, we need to substitute the given values of q, v, and B into the formula F = q(v x B) and solve for k.
Given:
- q = 5
- v = 3i - 2j + k
- B = i - 2j + k
Substituting these values into the formula, we have:
F = 5(3i - 2j + k) x (i - 2j + k)
Expanding the cross product, we get:
F = 5[(3i - 2j) x (i - 2j)] + 5[(3i - 2j) x k] + 5[k x (i - 2j)] + 5[k x k]
Using the properties of the cross product, we can simplify this expression:
F = 5[(3i - 2j) x (i - 2j)] + 5[(3i - 2j) x k] + 5[k x (i - 2j)] + 0
Since the cross product of a vector with itself is zero, the last term disappears.
Now, let's calculate each cross product:
(3i - 2j) x (i - 2j) = (3i - 2j) x i - (3i - 2j) x 2j
= (3i x i) - (2j x i) - (3i x 2j) + (2j x 2j)
= 0 - 2k - 6k + 0
= -8k
(3i - 2j) x k = (3i x k) - (2j x k)
= 3j - 2i
k x (i - 2j) = (k x i) - (k x 2j)
= -2i - k
Substituting these values back into the formula, we have:
F = 5(-8k) + 5(3j - 2i) + 5(-2i - k)
Simplifying further, we get:
F = -40k + 15j - 10i - 10i - 5k
= -20i + 15j - 45k
Therefore, k = -45.
To find the magnitude of the magnetic force F, we can use the formula |F| = |q(v x B)| = q|v x B|.
Substituting the given values, we have:
|F| = 5|v x B|
Since we have already calculated v x B as -20i + 15j - 45k, we can substitute this value:
|F| = 5|-20i + 15j - 45k|
Calculating the magnitude, we get:
|F| = 5 * sqrt((-20)^2 + 15^2 + (-45)^2)
= 5 * sqrt(400 + 225 + 2025)
= 5 * sqrt(2650)
Therefore, the magnitude of the magnetic force F is 5 * sqrt(2650).