Physics summer assignment vector multiplication

[grosenblatt]

Homework Statement

In the product (Vector F) = q(Vector v) X (Vector B), take q = 2,
(Vector v) = 2.0i + 4.0j + 6.0k
(Vector F) = 4.0i - 20j + 12k

What than is (Vector B) in unit-vector notation if Bx = By

Homework Equations

dot and cross products

The Attempt at a Solution

4.0i - 20j + 12k = (8_yb_z - 12_zb_y)i + (12_zb_x - 4_xb_z)j + (4_xb_y - 8_yb_x)k

 

[grosenblatt]

can anyone help? this is due tommorow =(. i really just don't know where to go from here at all

 

[baba89]

First, let's clarify the notation. In the homework statement, (Vector v) and (Vector B) are written in bold, indicating that they are vectors. However, in the attempt at a solution, they are written as scalars (2.0, 4.0, 6.0) and (Bx, By, Bz). For consistency, I will continue using vector notation throughout my response.

To solve this problem, we can use the properties of the cross product. First, we know that the cross product of two vectors is a vector that is perpendicular to both of the original vectors. This means that the vector (Vector F) is perpendicular to both (Vector v) and (Vector B).

Next, we can use the fact that the magnitude of the cross product is equal to the product of the magnitudes of the two original vectors multiplied by the sine of the angle between them. In this case, we know that the magnitude of (Vector F) is equal to 2 times the magnitude of (Vector v). We also know that the angle between (Vector v) and (Vector B) is 90 degrees, since they are perpendicular.

So, we can set up the equation:
|Vector F| = 2|Vector v|sin(90)

Solving for |Vector v|, we get:
|Vector v| = |Vector F|/2

Substituting in the values given in the homework statement, we get:
|Vector v| = √(4^2 + (-20)^2 + 12^2)/2 = 14

Now, we can use the fact that the cross product is equal to the product of the magnitudes of the two original vectors multiplied by the sine of the angle between them to find the magnitude of (Vector B).

|Vector B| = |Vector F|/(2|Vector v|)

Substituting in the values given in the homework statement, we get:
|Vector B| = √(4^2 + (-20)^2 + 12^2)/(2*14) = √(100/28) ≈ 2.380476

Finally, we can use the unit vector notation to express (Vector B). Since we know that the magnitude of (Vector B) is 2.380476, we can divide each component by this value to get the unit vector components.