Question Video: Identifying One of the Direction Angles of a Vector given the Other Two Mathematics

Suppose that 31°, 65°, and 𝜃 are the direction angles of a vector. Which of the following, to the nearest hundredth, is 𝜃? [A] 72.88° [B] 84.00° [C] 85.03° [D] 264.00°

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Video Transcript

Suppose that 31 degrees, 65 degrees, and 𝜃 are the direction angles of a vector. Which of the following, to the nearest hundredth, is 𝜃? Is it (A) 72.88 degrees, (B) 84.00 degrees, (C) 85.03 degrees, or (D) 264.00 degrees?

We recall that if the three direction angles of a vector are 𝛼, 𝛽, and 𝛾, then cos squared 𝛼 plus cos squared 𝛽 plus cos squared 𝛾 is equal to one. In this question, we will substitute 𝛼 equal to 31 degrees, 𝛽 equal to 65 degrees, and 𝛾 equal to 𝜃. This gives us the equation shown. We recall that cos squared of 31 degrees is the same as cos of 31 degrees all squared. This is how we type this into a scientific calculator. cos squared 31 degrees plus cos squared 65 degrees is equal to 0.91334 and so on. We can then subtract this from both sides of our equation, giving us cos squared 𝜃 is equal to 0.08665 and so on.

We then square root both sides of this equation. cos 𝜃 is equal to 0.29437 and so on. Finally, we take the inverse cos of both sides, such that 𝜃 is equal to 72.8797 and so on. We want the answer to the nearest hundredth. This means that the nine in the thousandths column is the deciding number. Rounding up gives us 72.88 degrees. Therefore, the correct answer is option (A). The three direction angles of the vector are 61 degrees, 65 degrees, and 72.88 degrees.

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