The diagram shows two vectors, 𝚨 and 𝚩. Each of the grid squares in the diagram has a side length of one. Calculate 𝚨 cross 𝚩.
Alright, so this is a question about vector products, and we’re presented with a diagram showing two vectors. We’re told that the grid squares in this diagram have a side length of one. And we’re asked to calculate the vector product 𝚨 cross 𝚩 of the two vectors in the diagram. Let’s start by writing out these two vectors in component form. We’ll need to find the 𝑥- and 𝑦-components of each vector from the diagram. We’ll add an 𝑥- and a 𝑦-axis to the diagram to make this process a little clearer. We see that vector 𝚨 extends positive four units in the 𝑥-direction and positive one unit in the 𝑦-direction.
Now, recall that the unit vector in the 𝑥-direction is labeled 𝐢 and the unit vector in the 𝑦-direction is 𝐣. So we can write the vector 𝚨 as its 𝑥-component, which is four, multiplied by 𝐢 plus its 𝑦-component, which is one, multiplied by 𝐣. For vector 𝚩, we see that it extends positive three units in the 𝑥-direction and positive five units in the 𝑦-direction. So we can write that 𝚩 equals its 𝑥-component, which is three, multiplied by 𝐢 plus its 𝑦-component, which is five, multiplied by 𝐣. We now have expressions for both vector 𝚨 and vector 𝚩 in component form.
Now, the question is asking us to calculate the vector product 𝚨 cross 𝚩. So let’s recall the definition of the vector product of two vectors. To do this, we’ll define two general vectors that lie in the 𝑥𝑦-plane. And we’ll label those vectors lowercase 𝐚 and 𝐛, where we use the lowercase letters to distinguish this general case from our two specific vectors from the question. We can write these general vectors in component form, labeling the 𝑥-components with a subscript 𝑥 and the 𝑦-components with a subscript 𝑦. Then, the vector product 𝐚 cross 𝐛 is defined as the 𝑥-component of 𝐚 multiplied by the 𝑦-component of 𝐛 minus the 𝑦-component of 𝐚 multiplied by the 𝑥-component of 𝐛 all multiplied by 𝐤, which is the unit vector in the 𝑧-direction.
We can use this definition in order to calculate the vector product of the two vectors from the question, capital 𝚨 and capital 𝚩. We are trying to calculate the vector product capital 𝚨 cross capital 𝚩. The first term is then the 𝑥-component of 𝚨, which is four, multiplied by the 𝑦-component of 𝚩, which is five. From this, we subtract the second term. This second term is the 𝑦-component of 𝚨, which is one, multiplied by the 𝑥-component of 𝚩, which is three. Then this whole thing is multiplied by the unit vector 𝐤. If we do the multiplications, the first term works out to 20 and the second term is three. Then, subtracting three from 20, we get our answer to the question that the vector product 𝚨 cross 𝚩 is equal to 17𝐤.