Given vectors $u =\langle 1, \sqrt{3}\rangle$ and $v =\langle 1,-1\rangle$, how are the vectors related?
A. The vectors form an acute angle of approximately $75^{\circ}$.
B. The vectors form an acute angle of approximately $81^{\circ}$.
C. The vectors form an obtuse angle of approximately $99^{\circ}$.
D. The vectors form an obtuse angle of approximately $105^{\circ}$.
Answer (2)
The vectors u = ⟨ 1 , 3 ⟩ and v = ⟨ 1 , − 1 ⟩ form an obtuse angle of approximately 10 5 ∘ . Thus, the correct option is D. The relationship is determined by calculating the dot product and the magnitudes of the vectors to compute the angle between them.
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Calculate the dot product of vectors u and v : $u
\cdot v = 1 - \sqrt{3}$.
Determine the magnitudes of u and v : ∣∣ u ∣∣ = 2 and ∣∣ v ∣∣ = 2 .
Compute the cosine of the angle between u and v : cos ( θ ) = 2 2 1 − 3 .
Find the angle θ using the inverse cosine function, resulting in an obtuse angle of approximately 10 5 ∘ .
Explanation
Problem Analysis We are given two vectors, u = ⟨ 1 , 3 ⟩ and v = ⟨ 1 , − 1 ⟩ , and we want to find the angle between them to determine their relationship.
Calculate the Dot Product First, we calculate the dot product of the two vectors: u ⋅ v = ( 1 ) ( 1 ) + ( 3 ) ( − 1 ) = 1 − 3
Calculate Magnitude of u Next, we find the magnitude of vector u :
∣∣ u ∣∣ = 1 2 + ( 3 ) 2 = 1 + 3 = 4 = 2
Calculate Magnitude of v Then, we find the magnitude of vector v :
∣∣ v ∣∣ = 1 2 + ( − 1 ) 2 = 1 + 1 = 2
Calculate Cosine of the Angle Now, we use the formula to find the cosine of the angle θ between the two vectors: cos ( θ ) = ∣∣ u ∣∣ ⋅ ∣∣ v ∣∣ u ⋅ v = 2 2 1 − 3
Calculate the Angle We calculate the angle θ by taking the inverse cosine: θ = arccos ( 2 2 1 − 3 ) Using a calculator, we find that: θ ≈ 10 5 ∘
Conclusion Since the angle between the vectors is approximately 10 5 ∘ , the vectors form an obtuse angle of approximately 10 5 ∘ .
Examples
Understanding the relationship between vectors is crucial in various fields. For instance, in physics, when analyzing forces acting on an object, vectors help determine the net force and direction. If two forces, represented by vectors, act at an angle, calculating the angle between them helps predict the object's motion. Similarly, in computer graphics, vectors are used to define the orientation and direction of objects in 3D space, and the angles between these vectors determine how objects interact with light and other objects.
