Let v = <3, -2>. What is the approximate direction angle of v?
A. 34°
B. 56°
C. 146°
D. 326°
Answer (2)
To find the direction angle of the vector v = ⟨ 3 , − 2 ⟩ , we will calculate the angle θ that the vector makes with the positive x-axis.
The formula to find the direction angle θ of a vector v = ⟨ a , b ⟩ is given by:
θ = tan − 1 ( a b )
In this case, a = 3 and b = − 2 . Plug these values into the formula:
θ = tan − 1 ( 3 − 2 )
Calculate tan − 1 ( 3 − 2 ) using a calculator to get an approximate result:
θ ≈ tan − 1 ( − 0.6667 ) ≈ − 33.6 9 ∘
Notice that this angle is measured counterclockwise from the positive x-axis. Since the angle is negative, it indicates that the direction is clockwise. We typically convert this to a positive angle by adding 360°:
θ = 36 0 ∘ + ( − 33.6 9 ∘ ) ≈ 326.3 1 ∘
Therefore, the approximate direction angle of the vector v = ⟨ 3 , − 2 ⟩ is about 32 6 ∘ .
The correct multiple-choice option is (D) 326° .
The direction angle of the vector v = ⟨ 3 , − 2 ⟩ is approximately 326° after calculating the angle made with the positive x-axis. The angle is obtained using the formula θ = tan − 1 ( a b ) and adjusted to a positive angle. Thus, the correct choice is D. 326° .
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