Find the angle between the vectors 𝐮 three, negative two and 𝐯 negative five, negative three. Give your answer to one decimal place.
In order to answer this question, we recall that the cosine of the angle between two vectors is equal to the dot product of two vectors 𝐮 and 𝐯 divided by the magnitude of vector 𝐮 multiplied by the magnitude of vector 𝐯. Let’s begin by calculating the dot product of vector 𝐮 and vector 𝐯.
We multiply the 𝑥- and 𝑦-components separately and then find the sum of these values. In this question, the 𝑥-components are three and negative five. The 𝑦-components are negative two and negative three. Multiplying a positive number and a negative number gives a negative answer. Therefore, three multiplied by negative five is negative 15. Multiplying two negative numbers in this case negative two and negative three gives us positive six. The dot product of vectors 𝐮 and 𝐯 is, therefore, equal to negative nine.
To find the magnitude of any vector, we find the sum of the squares of its individual components and then square root our answer. The magnitude of vector 𝐮 is equal to the square root of three squared plus negative two squared. Three squared is equal to nine. And negative two squared is equal to four. As nine plus four equals 13, the magnitude of vector 𝐮 is the square root of 13.
We repeat this to calculate the magnitude of vector 𝐯. Negative five squared is 25, and negative three squared is nine. These have a sum of 34. Therefore, the magnitude of vector 𝐯 is equal to the square root of 34.
Substituting in our values, we see that the cos of angle 𝜃 is equal to negative nine over the square root of 13 multiplied by the square root of 34. We can then calculate the angle 𝜃 by taking the inverse cosine of both sides of this equation. Typing this into our calculator, we get 𝜃 is equal to 115.346 and so on. We are asked to round our answer correct to one decimal place. As the deciding number in the hundredths column is less than five, we will round down. The angle between the two vectors 𝐮 and 𝐯 is equal to 115.3 degrees.