Given that vector 𝐀 is equal to 𝐢 plus eight 𝐣 minus three 𝐤 and vector 𝐁 is equal to nine 𝐢 plus 𝐣 plus five 𝐤, find the angle between the vectors 𝐀 and 𝐁, rounding your answer to the nearest degree.
In order to answer this question, we recall that if given two nonzero vectors 𝐮 and 𝐯, the cosine of the angle between them is equal to the dot product of 𝐮 and 𝐯 divided by the magnitude of vector 𝐮 multiplied by the magnitude of vector 𝐯. In this question, we are given two vectors 𝐀 and 𝐁 and we want to calculate the angle between them.
We begin by calculating the magnitude of each of our vectors. The magnitude of vector 𝐀 is equal to the square root of one squared plus eight squared plus negative three squared. We calculate the magnitude of any vector by square rooting the sum of the squares of the individual components. As one squared is equal to one, eight squared is 64, and negative three squared is nine, the magnitude of vector 𝐀 is the square root of 74. Using the same method, we see that the magnitude of vector 𝐁 is equal to the square root of nine squared plus one squared plus five squared. This is equal to the square root of 107.
We calculate the dot product of any two vectors by multiplying their corresponding components and then finding the sum of these values. 𝐀 dot 𝐁 is therefore equal to one multiplied by nine plus eight multiplied by one plus negative three multiplied by five. This is equal to two.
We can now substitute these three values into our formula to calculate the angle 𝜃. The cos of angle 𝜃 is equal to two divided by the square root of 74 multiplied by the square root of 107. We can then take the inverse cosine of both sides of this equation such that 𝜃 is equal to the inverse cos of two divided by root 74 multiplied by root 107. Typing the right-hand side into our calculator gives us 𝜃 is equal to 88.71 and so on.
We are asked to give our answer to the nearest degree. And as the number after the decimal point is greater than or equal to five, we round up. We can therefore conclude that the angle between the vectors 𝐀 and 𝐁 to the nearest degree is 89 degrees.