Question Video: Evaluating a Given Expression Using the Distributive Property of Dot Product

Video Transcript

If the magnitude of the vector 𝐀 is 10 and the magnitude of the vector 𝐁 is 17, where the size of the angle between 𝐀 and 𝐁 is 120 degrees, find the dot product of 𝐀 plus 𝐁 and 𝐀 minus two 𝐁 to the nearest hundredth.

We have two vectors, 𝐀 plus 𝐁 and 𝐀 minus two 𝐁, and we need to find their dot product. How can we do this? We can use the fact that the dot product is distributive over vector addition. So 𝐮 dot 𝐯 plus 𝐰 is equal to 𝐮 dot 𝐯 plus 𝐮 dot 𝐰, and 𝐮 plus 𝐯 dot 𝐰 is equal to 𝐮 dot 𝐰 plus 𝐯 dot 𝐰. So this works exactly the same as normal multiplication does over normal addition. So using the second of these rules with 𝐮 equal to 𝐀, 𝐯 equal to 𝐁, and 𝐰 equal to 𝐀 minus two 𝐁, we can expand the brackets giving us 𝐀 dot 𝐀 minus two 𝐁 plus 𝐁 dot 𝐀 minus two 𝐁.

And we can apply the first rule to expand the two terms that we’re left with. The first of these brackets becomes 𝐀 dot 𝐀 minus 𝐀 dot two 𝐁 and the second becomes 𝐁 dot 𝐀 minus 𝐁 dot two 𝐁. There is no more expanding to be done, so we can remove the distributive property from the screen.

And now we simplify each term. The dot product of any vector with itself is the magnitude of that vector squared. So 𝐀 dot 𝐀 becomes the magnitude of 𝐀 squared. The dot product of 𝐀 and two 𝐁 is just two times the dot product of 𝐀 and 𝐁. Why is this? Well, the dot product of 𝐀 and 𝐁 is the magnitude of 𝐀 times the magnitude of 𝐁 times the cosine of the angle 𝜃 between them.

How about the dot product of 𝐀 and two 𝐁? Notice that the angle between 𝐀 and two 𝐁 is the same as the angle between 𝐀 and 𝐁, namely 𝜃. The dot product of 𝐀 and two 𝐁 is therefore the magnitude of 𝐀 times the magnitude of two 𝐁 times the cosine of the same angle 𝜃. And as the magnitude of two 𝐁 is two times the magnitude of 𝐁, we get that this is two times the magnitude of 𝐀 times the magnitude of 𝐁 times cos 𝜃 which is two times the dot product of 𝐀 and 𝐁.

The dot product of 𝐁 and 𝐀 is equal to the dot product of 𝐀 and 𝐁. The dot product is commutative, and so it doesn’t matter which order the vectors are in. And we can see this is true because the dot product of 𝐁 and 𝐀 is the magnitude of 𝐁 times the magnitude of 𝐀 times the cosine of the angle between them. Normal real number multiplication is commutative, and so the magnitude of 𝐁 times the magnitude of 𝐀 is the same as the magnitude of 𝐀 times the magnitude of 𝐁. And so we can write this as the magnitude of 𝐀 times the magnitude of 𝐁 times cos 𝜃. And as the angle 𝜃 between 𝐁 and 𝐀 is the same as the angle 𝜃 between 𝐀 and 𝐁 — the order doesn’t matter, it’s the same angle — we see that 𝐁 dot 𝐀 is equal to 𝐀 dot 𝐁.

And for our fourth and final term, we use the same trick as in the second term to write 𝐁 dot two 𝐁 as two times 𝐁 dot 𝐁 which is of course two times the magnitude of 𝐁 squared.

We have some like terms that we can combine, and we can rewrite 𝐀 dot 𝐁 as the magnitude of 𝐀 times the magnitude of 𝐁 times the cosine of the angle 𝜃 between them. Now that we’ve simplified this expression as much as possible, we can use the values that we’re given in the question. The magnitude of 𝐀 is 10, the magnitude of 𝐁 is 17, and the measure of the angle 𝜃 between 𝐀 and 𝐁 is 120 degrees. We substitute these values in and either by using a calculator or doing it by hand, noticing that the cosine of 120 degrees is negative a half, we get an answer of negative 393.

We’re asked to give our answer to the nearest hundredth. Obviously, negative 393 to the nearest hundredth is just negative 393.00. So this is our answer to the required level of precision.