Consider the two vectors 𝐀 and 𝐁. 𝐀 has a length of four, and 𝐁 has a length of 13. The angle between them is 78 degrees. Calculate the magnitude of 𝐀 cross 𝐁. Give your answer to one decimal place.
Okay, so this is a question about vector products. We’re given two vectors 𝐀 and 𝐁, and we’re told the lengths of these two vectors. We’re also told the angle between them. We’re then asked to calculate the magnitude of the vector product 𝐀 cross 𝐁.
Let’s start by drawing a sketch of these two vectors. We can draw a vector 𝐀 pointing in some arbitrary direction and label its length as four units. Then vector 𝐁, with a length of 13 units, is at an angle of 78 degrees relative to vector 𝐀. The length of a vector is the magnitude of that vector. So we can say that the magnitude of vector 𝐀 is four and the magnitude of vector 𝐁 is 13.
We’re asked for the magnitude of the vector product 𝐀 cross 𝐁. Luckily for us, the magnitude of a vector product can be defined in terms of the quantities we are given in the question, namely, the magnitudes or lengths of the two vectors and the angle between those two vectors. Let’s define some general vectors, which we’ll label lowercase 𝐚 and lowercase 𝐛. And we’ll suppose that they have some angle 𝜃 between them. Then, the magnitude of the vector product 𝐚 cross 𝐛 is given by the magnitude of 𝐚 multiplied by the magnitude of 𝐛 multiplied by the sin of the angle 𝜃 between 𝐚 and 𝐛.
So in order to answer this question, we need to apply this general equation to our two vectors capital 𝐀 and capital 𝐁. We have that the magnitude of the vector product 𝐀 cross 𝐁 is given by the magnitude of 𝐀, which is four, multiplied by the magnitude of 𝐁, which is 13, multiplied by the sine of the angle between them, which is 78 degrees. If we calculate this, we get 50.863675 and so on with more decimal places.
The question specifies that we should give our answer to one decimal place. So the 50.86 et cetera rounds up to 50.9. And we have our answer to the question that the magnitude of the vector product 𝐀 cross 𝐁 to one decimal place is equal to 50.9.