Video Transcript
If the magnitude of vector 𝐀 is 17, the magnitude of vector 𝐁 is 12, and the dot product between vector 𝐀 and vector 𝐁 is 102, find the measure of the angle between the two vectors.
In this question, we’re given some information about two vectors. We’re told the magnitude of vector 𝐀 is 17, the magnitude of vector 𝐁 is 12, and the dot product between these two vectors is 102. We need to use this to determine the measure of the angle between these two vectors.
The first thing we might be worried about is the notation used for the dot product. This might not be the standard notation we’re used to. However, it means exactly the same thing. This represents the dot product or the scalar product of two vectors. So to answer this question, let’s start by recalling how we find the angle between two vectors. We recall if 𝜃 is the angle between two vectors, the vector 𝐮 and the vector 𝐯, then the dot product between vectors 𝐮 and 𝐯 will be equal to the magnitude of vector 𝐮 multiplied by the magnitude of vector 𝐯 times the cos of angle 𝜃.
There are a few things worth pointing out about this definition. First, when we say the angle between two vectors, we mean the angle between zero and 180 degrees. Next, we also assume that 𝐮 and 𝐯 are of the same dimension so that we can take the dot product between these two vectors. Finally, since the inverse cosine function has a range from zero to 180 degrees, we can use this to find the angle 𝜃 or, alternatively, the measure of this angle. All we would need to do is rearrange this to find an expression for 𝜃.
Therefore, if we set 𝜃 to be the angle between vector 𝐀 and vector 𝐁, we know the dot product between vector 𝐀 and vector 𝐁 will be equal to the magnitude of vector 𝐀 multiplied by the magnitude of vector 𝐯 [𝐁] multiplied by the cos of 𝜃. We’re told that the magnitude of vector 𝐀 is 17, the magnitude of vector 𝐁 is 12, and the dot product between vector 𝐀 and vector 𝐁 is 102. So we can substitute these into our expression to get 102 is equal to 17 multiplied by 12 times the cos of 𝜃.
We can then start simplifying this expression. First, 17 multiplied by 12 is equal to 204. We can then divide both sides of our equation through by 204 to get that 102 divided by 204 will be equal to the cos of 𝜃. We can then see both our numerator of 102 and our denominator of 204 share a factor of 102. So we can cancel this to get the fraction one-half. Finally, we can solve for angle 𝜃 by taking the inverse cosine of both sides of this equation. We get that 𝜃 is the inverse cos of one-half. And as we know the cos of 60 degrees is equal to one-half, we can conclude the inverse cos of one-half is equal to 60 degrees.
Therefore, we were able to show if the magnitude of vector 𝐀 is 17, the magnitude of vector 𝐁 is 12, and the dot product between vector 𝐀 and vector 𝐁 is 102, then the measure of the angle between these two vectors must be 60 degrees.
