If the cross product of vectors results in a vector of length 10, what is the angle between two vectors of lengths 5 and 3?

The cross product of two vectors is calculated using the formula:

[ | \mathbf{A} \times \mathbf{B} | = | \mathbf{A} | | \mathbf{B} | \sin(\theta) ]

where ( | \mathbf{A} | ) and ( | \mathbf{B} | ) are the magnitudes of the vectors and ( \theta ) is the angle between them. Given that the length of the cross product is 10, and the magnitudes of the vectors are 5 and 3, we can set up the equation:

[ 10 = 5 \cdot 3 \cdot \sin(\theta) ]

This simplifies to:

[ 10 = 15 \sin(\theta) ]

To isolate ( \sin(\theta) ), divide both sides by 15:

[ \sin(\theta) = \frac{10}{15} = \frac{2}{3} ]

Now, to find the angle ( \theta ), we need to determine the angle whose sine is ( \frac{2}{3} ).

When calculating the angle using a scientific calculator