Question Video: Calculating the Scalar Product of Two Vectors Using a Grid Physics

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.

We see over in our diagram our two vectors, 𝐀 and 𝐁, and that they’re laid out on a grid spacing. We’re told that each one of these grid squares has a side length of one. We’re not told the units of these lengths, but simply that the side lengths can be represented by one single unit, whatever our unit is. Knowing this, we want to calculate the scalar product of 𝐀 and 𝐁.

Now, we can recall that a scalar product involves combining two vectors. So, we’re off to a good start there because 𝐀 and 𝐁 are vectors. And we can recall further that, mathematically, the scalar product of two general vectors, 𝐀 and 𝐁, is equal to the product of their 𝑥-components plus the product of their 𝑦-components. Now, for our two specific vectors, also called 𝐀 and 𝐁, we don’t yet know their 𝑥- and 𝑦-components, but we can use this grid to find out.

We can start by laying down coordinate axes on this grid. Let’s say that for our origin, we pick the location where the tails of vectors 𝐀 and 𝐁 overlap. So, we’ll say that this is our 𝑥-axis, and this is our 𝑦. Relative to these axes, we can define the 𝑥- and 𝑦-components of our two vectors. Just as a side note, we could pick any orientation for our 𝑥- and 𝑦-axes so long as they’re perpendicular to one another and quantified vectors 𝐀 and 𝐁 that way. And our answer would come out the same.

Using these specific 𝑥- and 𝑦-axes though, let’s write out the components of vector 𝐀. We can see that along the 𝑥-axis, vector 𝐀 extends one, two, three units. So, that means vector 𝐀 is equal to three 𝐢, three units in the 𝑥-direction, plus some amount in the 𝑦-direction. Starting again at the origin, we count up one, two, three units and see that this is the vertical extent of vector 𝐀. Therefore, we can write vector 𝐀 as three 𝐢 plus three 𝐣. And now, we’ll do the same thing for vector 𝐁. The 𝑥-component of vector 𝐁 is equal to one, two, three, four, five, six units and its 𝑦-component is equal to one unit. And we can write that as one 𝐣 or simply 𝐣 so that vector 𝐁 overall is equal to six 𝐢 plus 𝐣.

Now that we know the components of our two vectors, we can use this relationship to solve for their scalar product. 𝐀 dot 𝐁 is equal to the 𝑥-component of vector 𝐀, we see that 𝑥-component is three, multiplied by the 𝑥-component of vector 𝐁. And we see that 𝑥-component is six. So, we have three times six. And to that, we add the 𝑦-component of vector 𝐀. That 𝑦-component is three multiplied by the 𝑦-component of vector 𝐁. And that 𝑦-component as we saw is one. So, 𝐀 dot 𝐁 is equal to three times six plus three times one, and that is equal to 18 plus three or 21. This is 𝐀 dot 𝐁, also called the scalar or dot product of 𝐀 and 𝐁.