1. The problem statement, all variables and given/known data
Find the vector product of Vector A cross Vector B(expressed in unit vectors) of the two vectors. What is the magnitude of the vector product?
2. Relevant equations
Vector A= 5.00i + 2.00j
Vector B= 3.00i - 1.00j
3. The attempt at a solution
I know how to get the dot product of the two, but how do I find the cross product??
The cross product is the determinant of a 3x3 matrix formed by the unit vectors i, j, k, A, and B. (Break A and B into x, y, and z components).
i j k
5 2 0
3 -1 0
The result is a vector orthogonal to both A and B; because A and B do not have z components, the result should be something like 0i + 0j + Ck, where C is a scalar.
To find the cross product of two vectors, Vector A and Vector B, you can use the following formula:
Vector A cross Vector B = (A_y * B_z - A_z * B_y)i + (A_z * B_x - A_x * B_z)j + (A_x * B_y - A_y * B_x)k
Where i, j, and k are the unit vectors in the x, y, and z directions, respectively.
Let's calculate the cross product of Vector A = 5.00i + 2.00j and Vector B = 3.00i - 1.00j:
Vector A cross Vector B = (2.00 * 0 - 0 * (-1.00))i + (0 * 3.00 - 5.00 * (-1.00))j + (5.00 * (-1.00) - 2.00 * 3.00)k
Simplifying this expression:
Vector A cross Vector B = 0i + (0 - (-5.00))j + (-5.00 - 6.00)k
Vector A cross Vector B = 0i + 5.00j - 11.00k
So, the cross product of Vector A and Vector B is 0i + 5.00j - 11.00k.
To find the magnitude of the cross product, you can use the Pythagorean theorem:
Magnitude = sqrt((0)^2 + (5.00)^2 + (-11.00)^2)
Simplifying this expression:
Magnitude = sqrt(0 + 25.00 + 121.00)
Magnitude = sqrt(146.00)
Magnitude ≈ 12.083
Therefore, the magnitude of the vector product is approximately 12.083.