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.
This question gives us two vectors 𝐀 and 𝐁 in the form of arrows drawn on a diagram. It then asks us to calculate the scalar product of these two vectors, 𝐀 dot 𝐁. Let’s begin by recalling the definition of the scalar product of two vectors. We’ll consider two general vectors, which we’ll label 𝐂 and 𝐃. And we’ll suppose that both of these vectors lie in the 𝑥𝑦-plane. Then, we can write these vectors 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. Then, the scalar product 𝐂 dot 𝐃 is equal to 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. So in general, the scalar product of two vectors is given by the product of the 𝑥-components of those two vectors plus the product of their 𝑦-components. This expression for the scalar product of two vectors tells us that if we want to calculate the scalar product 𝐀 dot 𝐁, then we’re going to 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 we’re told in the question that the grid squares in this diagram each have 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 count the number of squares that each vector extends in the 𝑥-direction and the 𝑦-direction. And since we know that each square 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 four units in the positive 𝑥-direction and two units in the positive 𝑦-direction. This means that the 𝑥-component of 𝐀 is four and the 𝑦-component is two. So we can write 𝐀 in component form as four 𝐢 hat plus two 𝐣 hat. Now we’ll do the same thing for vector 𝐁. We find that 𝐁 extends three units in the negative 𝑥-direction and six units in the positive 𝑦-direction. So the 𝑥-component of 𝐁 is negative three and the 𝑦-component is positive six. And in component form, we have that 𝐁 is equal to negative three 𝐢 hat plus six 𝐣 hat.
We now have both of our vectors 𝐀 and 𝐁 written in component form, which means that we’re 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 of this expression is given by the product of the 𝑥-components of the two vectors. So in our case, we need the 𝑥-component of 𝐀, which is four, multiplied by the 𝑥-component of 𝐁, which is negative three. Then we need to add to this the product of the 𝑦-components of the vectors. In this case, that’s the 𝑦-component of 𝐀, which is two, multiplied by the 𝑦-component of 𝐁, which is six.
We now have an expression for the scalar product 𝐀 dot 𝐁. And all that’s left to do is to evaluate this right-hand side. Our first term is four multiplied by negative three, and this gives us negative 12. Our second term is two multiplied by six, and this gives us positive 12. Then, we have negative 12 plus 12, which gives a result of zero. And so our final answer is that the scalar product 𝐀 dot 𝐁 is equal to zero.