Consider the two vectors 𝐫 of magnitude 12 and 𝐬 of magnitude 26. The angle 𝜃 between them is 68 degrees. What is the scalar product of 𝐫 and 𝐬? Give your answer to the nearest integer.
Okay, so in this question, we are given two vectors, 𝐫 and 𝐬. We are told the magnitude of these two vectors, and we’re also given the angle between them. We are asked to find the scalar product of these two vectors. Let’s begin by drawing out the two vectors just to get a sense of what’s going on. Vector 𝐫 has a magnitude of 12. We don’t know anything about the direction of 𝐫 in absolute terms, only relative to our second vector 𝐬. We’ll see that the absolute direction won’t actually matter. So let’s draw 𝐫 pointing upwards. Vector 𝐬 is at an angle of 68 degrees to vector 𝐫 and has a magnitude of 26. This is just over twice the magnitude of 𝐫. So the arrow representing 𝐬 is just over twice the length of that for 𝐫.
To calculate the scalar product of these two vectors, we need to recall the geometric formula for the scalar product. For two general vectors 𝐀 and 𝐁, the scalar product of those two vectors is defined as the magnitude of 𝐀 multiplied by the magnitude of 𝐁 multiplied by the cos of the angle 𝜃 between them. And of course, we can write this same formula for our two vectors 𝐫 and 𝐬. We know from the question that the magnitude of 𝐫 is 12, and we know that the magnitude of 𝐬 is 26. Finally, we know that the angle between these two vectors is 68 degrees. So let’s put in these numbers into our equation for the scalar product.
So we have that the scalar product 𝐫 dot 𝐬 is given by 12, that’s the magnitude of 𝐫, multiplied by 26, the magnitude of 𝐬, multiplied by the cos of 68 degrees, that’s the angle 𝜃 between the vectors 𝐫 and 𝐬. When we do this calculation, we find that the result is 116.877 and so on with further decimal places. But if we look back to the question, we see that we are told to give our answer to the nearest integer. So if we round this number to the nearest integer, we see that it rounds up to 117. And so we get our answer to the question that to the nearest integer, the scalar product of 𝐫 and 𝐬 is 117.