Question Video: Finding the Dot Product between Vectors Mathematics • Third Year of Secondary School

Video Transcript

The angle between vector 𝐀 and vector 𝐁 is 22 degrees. If the magnitude of vector 𝐀 is equal to three times the magnitude of vector 𝐁 is equal to 25.2, find the dot product between 𝐀 and 𝐁 to the nearest hundredth.

In this question, we’re given some information about two vectors 𝐀 and 𝐁. First, we’re told the angle between these two vectors is equal to 22 degrees. Next, we’re also told information about their magnitudes. We know the magnitude of 𝐀 is equal to 25.2, and we know that three times the magnitude of 𝐁 is also equal to 25.2. So the magnitude of 𝐀 is three times bigger than the magnitude of 𝐁. We need to use this to find the dot product of 𝐀 and 𝐁. And we need to give our answer to the nearest hundredth.

To answer this question, we need to notice that we know a formula which connects the angle between two vectors with their dot product. We recall if 𝜃 is the angle between two vectors 𝐀 and 𝐁, then we know that the cos of 𝜃 must be equal to the dot product between 𝐀 and 𝐁 divided by the magnitude of 𝐀 times the magnitude of 𝐁. And in this question, we already know some of these values. For example, we’re told the angle between our two vectors is 22 degrees. Next, we’re also told the magnitude of 𝐀 is equal to 25.2.

And we could also find the magnitude of 𝐁 using the information given to us in the question. One way of doing this is to notice that three times the magnitude of 𝐁 is equal to 25.2. We can then solve this to find the magnitude of 𝐁 by dividing both sides of our equation through by three. And calculating this, we get the magnitude of 𝐁 is 8.4. So, in fact, the only unknown in this equation is the dot product between 𝐀 and 𝐁. And that’s exactly what we’re asked to calculate.

So we’ll substitute the angle of 𝜃 equal to 22 degrees, the magnitude of 𝐀 equal to 25.2, and the magnitude of 𝐁 equal to 8.4 into our equation. This gives us the cos of 22 degrees should be equal to the dot product between 𝐀 and 𝐁 divided by 25.2 times 8.4. And now we can just rearrange this equation for the dot product between 𝐀 and 𝐁. We multiply through by 25.2 times 8.4. This gives us 𝐀 dot 𝐁 is 25.2 times 8.4 times the cos of 22 degrees. And we can calculate this to the nearest hundredth or to two decimal places. It’s equal to 196.27.