Given that vector 𝐀 is equal to four 𝐢 minus 𝐣 minus two 𝐤 and vector 𝐁 equals two, negative two, four, determine, to the nearest hundredth, the measure of the smaller angle between the two vectors.
The first point to note in this question is that our two vectors have been written using different notation. Vector 𝐀 is equal to four 𝐢 minus 𝐣 minus two 𝐤. Vector 𝐁 can be rewritten in the same form as two 𝐢 minus two 𝐣 plus four 𝐤. We’ll recall at this stage that the cosine of the angle between two vectors 𝜃 is equal to the dot product of the two vectors divided by the product of the magnitude of the two vectors. We calculate the dot product of the two vectors 𝐀 and 𝐁 by multiplying the coefficients of 𝐢, the coefficients of 𝐣, and the coefficients of 𝐤. We then work out the sum of these three answers.
Four multiplied by two is equal to eight. Negative one multiplied by negative two is equal to positive two. And negative two multiplied by four is equal to negative eight. Adding this is the same as subtracting eight. Eight plus two is equal to 10 and subtracting eight gives us two. This is the value of the dot product. We calculate the magnitude of any vector using the formula the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared, where 𝑥, 𝑦, and 𝑧 are the coefficients of 𝐢, 𝐣, and 𝐤, respectively.
The magnitude of vector 𝐀 in this question is equal to the square root of four squared plus negative one squared plus negative two squared. This is equal to the square root of 21. In the same way, the magnitude of vector 𝐁 is equal to two squared plus negative two squared plus four squared. This is equal to the square root of 24. We can now substitute our three values into the formula.
The cosine of 𝜃 is equal to two divided by the square root of 21 multiplied by the square root of 24. We can then calculate the angle 𝜃 by taking the inverse cosine or cosine to the minus one of two divided by the square root of 21 multiplied by the square root of 24. This is equal to 84.8889 and so on. We were asked to round to the nearest hundredth. This is the same as rounding to two decimal places. As eight is greater than five, we round up.
The measure of the angle between the two vectors is 84.89 degrees.