Video: Calculating the Scalar Product of Two Vectors Shown on a Grid

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

04:38

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.

This question gives us two vectors, 𝐀 and 𝐁, in the form of arrows drawn on a diagram. We are asked to find 𝐀 dot 𝐁, the scalar product of these two vectors. So 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. If we suppose that both of these vectors lie in the π‘₯𝑦-plane, then we can write them in component form as an π‘₯-component labeled with a subscript π‘₯ multiplied by 𝐒 hat plus a 𝑦-component labeled with a subscript 𝑦 multiplied by 𝐣 hat. Remember that 𝐒 hat is the unit vector in the π‘₯-direction and 𝐣 hat is the unit vector in the 𝑦-direction.

Then, the scalar product 𝐂 dot 𝐃 is equal to the π‘₯-component of 𝐂 multiplied by the π‘₯-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 their π‘₯-components plus the product of their 𝑦-components. This general expression for the scalar product of two vectors tells us that if we want to work out the scalar product 𝐀 dot 𝐁, then we’re going to need to find the π‘₯- and 𝑦-components of our vectors 𝐀 and 𝐁.

Our vectors 𝐀 and 𝐁 are given as arrows drawn on a diagram. And the question tells us that each of the grid 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 π‘₯-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 us the π‘₯- and 𝑦-components of our vectors.

Let’s begin by counting the squares for vector 𝐀. We see that 𝐀 extends four squares in the negative π‘₯-direction and two squares in the negative 𝑦-direction. So the π‘₯-component of 𝐀 is negative four, and the 𝑦-component is negative two. This means we can write the vector 𝐀 in component form as negative four 𝐒 hat minus two 𝐣 hat. Now, we’ll do the same thing for vector 𝐁. We see that 𝐁 extends three squares in the negative π‘₯-direction and three squares in the negative 𝑦-direction. So the π‘₯-component of 𝐁 is negative three and the 𝑦-component of 𝐁 is also negative three.

Then in component form, we can write the vector 𝐁 as negative three 𝐒 hat minus three 𝐣 hat. We now have both our vectors 𝐀 and 𝐁 written in component form, which means that we are 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 given by the product of the π‘₯-components of the two vectors. So for the scalar product 𝐀 dot 𝐁, that’s the π‘₯-component of 𝐀, which is negative four, multiplied by the π‘₯-component of 𝐁, which is negative three.

Then we add a second term to this given by the product of the 𝑦-components of the vectors. So for us, that’s the 𝑦-component of 𝐀, which is negative two, multiplied by the 𝑦-component of 𝐁, which is negative three. The final step is then to evaluate this expression here. The first term is negative four multiplied by negative three, which gives positive 12. The second term is negative two multiplied by negative three, which gives positive six. Then 12 plus six gives us a result of 18. And so our answer to the question is that the scalar product 𝐀 dot 𝐁 is equal to 18.

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