Find the angle between the vector 𝚨 which is equal to one, negative eight, eight and the unit vector 𝐤. Round your answer to the nearest degree.
We begin by recalling that the unit vectors, denoted 𝐢 hat, 𝐣 hat, and 𝐤 hat, are the vectors of magnitude one in the 𝑥-, 𝑦-, and 𝑧-direction, respectively. The unit vector 𝐤 therefore has a displacement of zero in the 𝑥-direction, zero in the 𝑦-direction, and one in the 𝑧-direction. This vector is equal to zero, zero, one.
We are also told in this question that vector 𝚨 is equal to one, negative eight, eight. And we need to calculate the angle between them. We recall that if we’re given two nonzero vectors 𝐮 and 𝐯, the cos of the angle 𝜃 between them is equal to the dot product of vector 𝐮 and vector 𝐯 divided by the magnitude of vector 𝐮 multiplied by the magnitude of vector 𝐯. The magnitude of the vector zero, zero, one is equal to one. And we will call this vector vector 𝚩.
The magnitude of vector 𝚨 is equal to the square root of one squared plus negative eight squared plus eight squared. We find the sum of the squares of the individual components and then square root the answer to calculate the magnitude. The magnitude of vector 𝚨 is therefore equal to the square root of 129.
To calculate the dot product of any two vectors, we multiply the corresponding components and then find the sum of these values. 𝚨 dot 𝚩 is equal to one multiplied by zero plus negative eight multiplied by zero plus eight multiplied by one. This is equal to eight. We can now substitute these three values into our formula to calculate the angle 𝜃.
The cos of angle 𝜃 is equal to eight divided by the square root of 129 multiplied by one. Taking the inverse cos of both sides of this equation, we have 𝜃 is equal to the inverse cos of eight over the square root of 129. Typing the right-hand side into our calculator gives us 45.22 and so on. We are asked to round our answer to the nearest degree. Therefore, 𝜃 is equal to 45 degrees. The angle between the vector one, negative eight, eight and the unit vector 𝐤 is 45 degrees.