Question Video: Calculating the Cross Product of Two Vectors Given on a Grid Physics

The diagram shows two vectors, 𝐀 and 𝐁. Each of the grid squares in the diagram has a side length of 1. Calculate 𝐀 Γ— 𝐁.

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Video Transcript

The diagram shows two vectors, 𝐀 and 𝐁. Each of the grid squares in the diagram has a side length of one. Calculate 𝐀 cross 𝐁.

Right, so this is a question about vector products. We have a diagram showing two vectors labeled 𝐀 and 𝐁. We are told that the grid squares have sides of length one. And then we’re asked to calculate the vector product 𝐀 cross 𝐁 of these two vectors.

Let’s start by writing out the vectors in component form. To do this, we need to find the π‘₯- and 𝑦-components of each vector from the diagram. Let’s add an π‘₯- and a 𝑦-axis to the diagram to make this process a little clearer. We see that vector 𝐀 extends positive four units in the π‘₯-direction and negative four units in the 𝑦-direction. If we recall that 𝐒 is the unit vector in the π‘₯-direction and 𝐣 is the unit vector in the 𝑦-direction, then we can write that the vector 𝐀 is equal to its π‘₯-component four multiplied by 𝐒 plus its 𝑦-component negative four multiplied by 𝐣. Or more simply, we could write this as four 𝐒 minus four 𝐣.

For vector 𝐁, we see that it extends negative five units in the π‘₯-direction and negative one unit in the 𝑦-direction. So we can write that 𝐁 equals its π‘₯-component negative five multiplied by 𝐒 plus its 𝑦-component negative one multiplied by 𝐣, which we could also write negative five 𝐒 minus 𝐣.

So now we have expressions for both 𝐀 and 𝐁 in component form. The question is asking us to calculate the vector product 𝐀 cross 𝐁. So let’s recall the definition of the vector product of two vectors. Let’s define two general vectors that lie in the π‘₯𝑦-plane and label these lowercase 𝐚 and lowercase 𝐛. Well, we’ve used 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 π‘₯-components with a subscript π‘₯ and the 𝑦-components with a subscript 𝑦. Then, the vector product 𝐚 cross 𝐛 is defined as the π‘₯-component of 𝐚 multiplied by the 𝑦-component of 𝐛 minus the 𝑦-component of 𝐚 multiplied by the π‘₯-component of 𝐛 all multiplied by 𝐀, which is the unit vector in the 𝑧-direction.

We can use this definition to calculate the vector product of our two vectors from the question as capital 𝐀 and capital 𝐁. It’s going to be important here to keep track of all the negative signs in the components as we go. We are asked to calculate 𝐀 cross 𝐁. So the first term is the π‘₯-component of 𝐀, which is four, multiplied by the 𝑦-component of 𝐁, which is negative one. Then, from this, we subtract the second term. This second term is the 𝑦-component of 𝐀, which is negative four, multiplied by the π‘₯-component of 𝐁, which is negative five. Then, this whole thing is multiplied by the unit vector 𝐀.

This first term, four multiplied by negative one, gives us negative four. The second term, negative four multiplied by negative five, gives us positive 20. But remember that we are subtracting this second term from the first. So we have negative four minus 20 all multiplied by 𝐀. Subtracting 20 from negative four, we get the answer to the question that the vector product 𝐀 cross 𝐁 is equal to negative 24𝐀.

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