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 dot 𝐁.
In this question, we’re given two vectors 𝐀 and 𝐁 in the form of arrows drawn on a diagram. We are asked to work out the scalar product of these two vectors, 𝐀 dot 𝐁. So let’s start by recalling the definition of the scalar product of two vectors. We’ll consider two general vectors, which we’ll label 𝐂 and 𝐃. Supposing that both of these vectors lie in the 𝑥𝑦-plane, then we can write them in component form as an 𝑥-component labeled with a subscript 𝑥 multiplied by 𝐢 hat plus a 𝑦-component labeled with a subscript 𝑦 multiplied by 𝐣 hat. Remember that 𝐢 hat is the unit vector in the 𝑥-direction and 𝐣 hat is the unit vector in the 𝑦-direction.
The scalar product 𝐂 dot 𝐃 is then defined as the 𝑥-component of 𝐂 multiplied by the 𝑥-component of 𝐃 plus the 𝑦-component of 𝐂 multiplied by the 𝑦-component of 𝐃. In other words, the scalar product of two vectors is given by the product of the vectors’ 𝑥-components plus the product of their 𝑦-components. Looking at this general expression for the scalar product of two vectors, we see that if we want to calculate the scalar product 𝐀 dot 𝐁, then we need to work out the 𝑥- and 𝑦-components of our vectors 𝐀 and 𝐁.
Now, the vectors 𝐀 and 𝐁 are given to us as arrows drawn on a diagram, and the question tells us that each of the squares in this diagram has a side length of one. If we add a set of axes to our diagram with the origin positioned at the tail of the two vectors, then we can easily read off the number of squares that each vector extends in the 𝑥-direction and in the 𝑦-direction. Since we know that each of these squares has a side length of one, then the number of squares directly gives the 𝑥- and 𝑦-components of the vectors.
Let’s begin by counting the squares for vector 𝐀. We see that 𝐀 extends two squares in the positive 𝑥-direction and three squares in the positive 𝑦-direction. So the 𝑥-component of 𝐀 is two and the 𝑦-component is three. And we can write the vector 𝐀 in component form as two 𝐢 hat plus three 𝐣 hat. Now we’ll do the same thing with vector 𝐁. We find that 𝐁 extends four squares in the positive 𝑥-direction and four squares in the negative 𝑦-direction. So the vector 𝐁 has an 𝑥-component of four and a 𝑦-component of negative four. Then, we can write 𝐁 in component form as four 𝐢 hat minus for 𝐣 hat.
So we now have expressions for both vectors 𝐀 and 𝐁 in component form, which means that we are now ready to calculate the scalar product 𝐀 dot 𝐁. From our general expression for the scalar product of two vectors, we see that the first term is the product of the 𝑥-components of the vectors. So for our scalar product 𝐀 dot 𝐁, that’s the 𝑥-component of 𝐀, which is two, multiplied by the 𝑥-component of 𝐁, which is four. Then we add to this a second term given by the product of the 𝑦-components of the vectors. For us, that’s the 𝑦-component of 𝐀, which is three, multiplied by the 𝑦-component of 𝐁, which is negative four.
The last step left to go is then to evaluate this expression here. The first term is two multiplied by four, which gives us eight. And the second term is three multiplied by negative four, which gives us negative 12. Then, we have eight plus negative 12, which gives us negative four. And so our answer to the question is that the scalar product 𝐀 dot 𝐁 is equal to negative four.
