In the product F(vector)=q(V(vector)xB(vector), take q = 4,
V(vector)= 2.0i + 4.0j + 6.0k and
F(vector)= 136i -176j + 72k.
What then is B(vector) in unit-vector notation if
Bx = By?
To find B(vector) in unit-vector notation, we can first use the formula for the cross product of two vectors:
F(vector) = q(V(vector) x B(vector))
where x represents the cross product operation.
Given that F(vector) = 136i -176j + 72k, q = 4, and V(vector) = 2.0i + 4.0j + 6.0k, we can substitute these values into the formula:
136i - 176j + 72k = 4((2.0i + 4.0j + 6.0k) x B(vector))
Next, we expand the cross product term using the properties of cross products:
(2.0i + 4.0j + 6.0k) x B(vector) = (4.0k - 6.0j)Bx + (-6.0k + 2.0i)By + (2.0j - 4.0i)Bz
Comparing the coefficients of i, j, and k on both sides of the equation, we get:
136 = -12By - 4Bz
-176 = -6Bx + 4Bz
72 = 2Bx + 6By
Since we are given that Bx = By, we substitute this into the last equation:
72 = 2Bx + 6Bx
72 = 8Bx
Bx = 9
Substituting this value of Bx into the second equation gives us:
-176 = -6(9) + 4Bz
-176 = -54 + 4Bz
4Bz = -176 + 54
4Bz = -122
Bz = -30.5
Finally, we substitute the values of Bx and Bz into the first equation:
136 = -12By - 4(-30.5)
136 = -12By + 122
12By = -122 + 136
12By = 14
By = 1.17
Therefore, B(vector) in unit-vector notation is B(vector) = 9i + 1.17j - 30.5k.