Calculate the product 𝐂 cross 𝐅.
We see in the diagram that 𝐂 and 𝐅 are both vectors. 𝐂 has a magnitude of 8.0 and is inclined at an angle called 𝛾 of 30 degrees from the positive horizontal direction. 𝐅 has a magnitude of 20.0 and is inclined at an angle called 𝜑 of 110 degrees from that same axis.
We want to solve for the cross product of these two vectors. And to do that, we want to break them up into their two component parts. If we call the direction to the right the positive 𝑖 hat direction and up the positive 𝑗 hat direction, then starting with 𝐂, we can write it in component form this way: 𝐂 is equal to the vectors magnitude 8.0 multiplied by the cosine of 30 degrees in the 𝑖 hat direction plus the sine of 30 degrees in the 𝑗 hat direction. We can write the components of 𝐅 in a similar way. It’s equal to the magnitude of the vector 20.0 times the cosine of 110 degrees in the 𝑖 hat direction plus the sine of 110 degrees in the 𝑗 hat.
Now that we know 𝐂 and 𝐅 according to their components, we can recall how to perform a cross product operation. The cross product of two vectors — we can call them 𝐀 and 𝐁 — is equal to the determinant of this matrix, where the columns are divided up into the 𝑖, 𝑗, and 𝑘 unit vector directions and the last two rows are those respective components of 𝐀 and 𝐁.
Applying this relationship to our vectors 𝐂 and 𝐅, we write them out in this matrix and include their 𝑖, 𝑗, and 𝑘 components for 𝐂 and then for 𝐅. Because the 𝑘 components of both 𝐂 and 𝐅 are zero, that means according to the cross product that the 𝑖 and 𝑗 components on this cross product will also be zero. In other words, we only need to calculate the 𝑘 component of this cross product because that’s all that will be left — that’s all that will be nonzero.
When we calculate this 𝑘 component, we see it’s equal to 8.0 times the cosine of 30 multiplied by 20.0 times the sine of 110 degrees minus 8.0 times the sine of 30 degrees times 20.0 times the cosine of 110 degrees.
When we enter this expression on our calculator, we find a result of 158𝑘. So the cross product of these two vectors 𝐂 and 𝐅 is entirely in the 𝑘 direction, pointing out of the page as we view it with the magnitude of 158 units.