If 𝐀 is the vector two 𝐢 and 𝐁 is the vector 𝐢 minus nine 𝐣, calculate the scalar product of 𝐀 and 𝐁.
In this question, we’re given two vectors, the vector 𝐀 and the vector 𝐁, in terms of the unit directional vectors 𝐢 and 𝐣. We’re asked to calculate the scalar product of vector 𝐀 and vector 𝐁. To do this, let’s start by recalling exactly what we mean by the scalar product. This is sometimes referred to as the dot product of two vectors because we represent it by a dot. In either case, it’s an operation between two vectors which outputs a scalar. And to calculate this, we need to multiply the corresponding components of our vectors together and then add all of these products together.
So in terms of the unit directional vectors, this gives us that the dot product between 𝑎𝐢 plus 𝑏𝐣 and 𝑐𝐢 plus 𝑑𝐣 will be 𝑎𝑐 plus 𝑏𝑑. We multiply the first component of each vector together; that’s 𝑎 multiplied by 𝑐. And then we add to this the product of the second components together. That’s 𝑏 multiplied by 𝑑. We’re now ready to find the dot product between the two vectors given to us in the question. That’s the dot product between two 𝐢 and 𝐢 minus nine 𝐣. To do this, it might be worth noticing that we can rewrite the vector two 𝐢 as to 𝐢 plus zero 𝐣.
Now to calculate the dot product between these two vectors, we need to multiply the corresponding components together and then add the results. Multiplying the first component of each vector together and remembering the coefficient of 𝐢 is going to be one gives us two times one. And remembering our second vector has a coefficient of 𝐣 equal to zero, multiplying these together, we get zero multiplied by negative nine. And of course we can calculate this. It’s equal to two. Therefore, we were able to show if 𝐀 is the vector two 𝐢 and 𝐁 is the vector 𝐢 minus nine 𝐣, then the scalar product between 𝐀 and 𝐁 will be equal to two.