Given a vector A = 4.6i + 4.6j and a vector B = −4.6i + 4.6j, determine the magnitude of the cross product of these two vectors, A ⨯ B.
Hint: Make a sketch of both vectors including a coordinate system.
To determine the magnitude of the cross product of two vectors, we follow these steps:
Step 1: Calculate the cross product of the two vectors.
The cross product of two vectors, A and B, is given by the formula A ⨯ B = |A| |B| sin(θ) n, where |A| and |B| are the magnitudes of vectors A and B, θ is the angle between the two vectors, and n is the unit vector perpendicular to the plane formed by A and B.
In this case, vector A = 4.6i + 4.6j and vector B = -4.6i + 4.6j.
To calculate the cross product, we use the determinant of a 3x3 matrix. Let's create a matrix using the i, j, and k unit vectors and the components of vectors A and B:
| i j k |
| 4.6 4.6 0 |
|-4.6 4.6 0 |
Taking the determinant of this matrix:
A ⨯ B = (4.6 * 4.6) - (-4.6 * 4.6) = 46
Step 2: Calculate the magnitude of the cross product.
The magnitude of the cross product of two vectors is given by the formula |A ⨯ B| = |A| |B| sin(θ), where θ is the angle between the two vectors.
Since the vectors A and B have the same magnitude, |A| = |B| = 6.48 (applying the Pythagorean theorem).
To find the angle θ, we can use the dot product formula: A · B = |A| |B| cos(θ).
Calculating the dot product of A and B:
A · B = (4.6 * -4.6) + (4.6 * 4.6) = 0
Using the dot product formula to find the angle:
0 = 6.48 * 6.48 * cos(θ)
cos(θ) = 0
θ = 90 degrees
Now, substituting the values into the formula for the magnitude of the cross product:
|A ⨯ B| = |A| |B| sin(θ) = 6.48 * 6.48 * sin(90 degrees) = 46.85
Therefore, the magnitude of the cross product of vectors A and B, A ⨯ B, is approximately 46.85.