To find the magnitude and direction of the cross product of two vectors, we can denote these vectors as **A** and **B**. The cross product, represented as **C = A × B**, results in a new vector **C** that is perpendicular to both **A** and **B**. The magnitude of the cross product can be calculated using the formula:
|C| = |A| |B| \sin(\theta)
In this equation, |A| and |B| are the magnitudes of vectors **A** and **B**, respectively, and θ is the angle between them. For example, if |A| = 12 and |B| = 8, we need to determine the sine of the angle between them. In a three-dimensional context, it can be tricky to visualize the angle. If **A** is horizontal and **B** is vertical, the angle between them is actually 90 degrees, even if it appears otherwise in a diagram.
Using the sine of 90 degrees, which is 1, we can calculate the magnitude of the cross product:
|C| = 12 × 8 × 1 = 96
Next, to determine the direction of vector **C**, we apply the right-hand rule. This involves pointing the fingers of your right hand in the direction of vector **A** and curling them towards vector **B**. The direction your thumb points indicates the direction of vector **C**. In this case, if **A** is tilted away from you and **B** is pointing straight up, your thumb will point off to the right, indicating the direction of **C**.
To find the angle θ that vector **C** makes with the x-axis, we can analyze the situation from a top-down perspective. If we know that the angle between the x-axis and vector **A** is 30 degrees, and since vector **C** is perpendicular to both **A** and **B**, we can deduce that the angle between the x-axis and vector **C** is also 30 degrees. Thus, the positive angle from the x-axis to vector **C** is:
θ = 30 degrees
In summary, the magnitude of the cross product **C** is 96, and it makes a positive angle of 30 degrees with the x-axis.
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