Question Video: Finding the Angle between Two Given Vectors Mathematics • Third Year of Secondary School

Video Transcript

Find the angle 𝜃 between the vectors 𝐕 seven, two, negative 10 and the vector 𝐖 two, six, four. Give your answer correct to one decimal place.

In this question, we’re asked to find the angle 𝜃 between two vectors: the vector 𝐕 seven, two, negative 10 and the vector 𝐖 two, six, four. We need to give our answer correct to one decimal place. 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 𝚨 and the vector 𝚩, then the cos of 𝜃 will be equal to the dot product between vector 𝚨 and vector 𝚩 divided by the norm of vector 𝚨 multiplied by the norm of vector 𝚩. It’s also worth pointing out the same is true in reverse. If 𝜃 satisfies this equation, then 𝜃 can be said to be the angle between the two vectors 𝚨 and 𝚩.

As a convention, when we say the angle between two vectors, we normally mean the smallest nonnegative value of 𝜃. We can find this by just taking the inverse cosine of both sides of the equation. Therefore, to find the value of 𝜃 given to us in this question, we need to find the dot product between our two vectors, 𝐕 and 𝐖, and we need to find the norms of both vector 𝐕 and vector 𝐖. Let’s start by finding the dot product. We need to find the dot product between the vectors 𝐕 and 𝐖. That’s the dot product between the vector seven, two, negative 10 and the vector two, six, four.

Remember, to find the dot product between two vectors, we need to find the product of the corresponding components of the two vectors and then add the results together. In our case, we get the dot product between vector 𝐕 and 𝐖 is seven times two plus two times six plus negative 10 multiplied by four. And if we calculate this expression, we see it’s equal to negative 14.

Next, we’re going to need to calculate the norm of vector 𝐕 and the norm of vector 𝐖. To do this, we recall the norm of a vector is equal to the positive square root of the sum of the squares of its components. In other words, the norm of the vector 𝐚, 𝐛, 𝐜 will be equal to the square root of 𝑎 squared plus 𝑏 squared plus 𝑐 squared. We can use this to find the norm of vectors 𝐕 and 𝐖. Let’s start with the norm of vector 𝐕. The square roots of the sum of the squares of the components of vector 𝐕 is the square root of seven squared plus two squared plus negative 10 all squared. And if we calculate this, we see that the norm of vector 𝐕 is equal to the square root of 153.

We can do exactly the same for vector 𝐖. The norm of vector 𝐖 will be the square root of the sum of the squares of its components. That’s the square root of two squared plus six squared plus four squared. And if we calculate this expression, we see that the norm of vector 𝐖 is root 56. We’re now ready to substitute these values into our formula involving 𝜃, the angle between vectors 𝐕 and 𝐖. We know if 𝜃 is the angle between vectors 𝐕 and 𝐖, the cos of 𝜃 will be equal to the dot product between vector 𝐕 and 𝐖 divided by the norm of vector 𝐕 multiplied by the norm of vector 𝐖.

We already calculated the dot product between vector 𝐕 and vector 𝐖. We found this was equal to negative 14. Similarly, we also found the norm of vector 𝐕 to be root 153 and the norm of vector 𝐖 to be root 56. Therefore, the cos of angle 𝜃 will be equal to negative 14 divided by root 153 multiplied by root 56. We could simplify this expression. However, it’s not necessary. We only need to find the value of 𝜃. And to do this we need to take the inverse cosine of both sides of this equation. We see that 𝜃 will be the inverse cos of negative 14 divided by 153 times root 56.

We can then use our calculator to evaluate this expression. We’re not told whether to use degrees or radians in this question, so we’ll just use degrees. We get 98.699 and this expansion continues degrees. The question does want us to give our answer to one decimal place. To do this, we notice the second decimal place in our expansion is nine. This means we’re going to need to round up. And this gives us our final answer. The angle 𝜃 between the two vectors 𝐕 seven, two, negative 10 and 𝐖 two, six, four to one decimal place is 98.7 degrees.