Consider vectors [tex]u[/tex] and [tex]v[/tex], where [tex]u =\langle 1,-1\rangle[/tex] and [tex]v =\langle 1,1\rangle[/tex]. What is the measure of the angle between the vectors?
A. [tex]0^{\circ}[/tex]
B. [tex]60^{\circ}[/tex]
C. [tex]90^{\circ}[/tex]
D. [tex]180^{\circ}[/tex]
Answer (2)
The angle between the vectors u = ⟨ 1 , − 1 ⟩ and v = ⟨ 1 , 1 ⟩ is 9 0 ∘ . Therefore, the correct answer is C.
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Calculate the dot product of the vectors u and v : u ⋅ v = 0 .
Compute the magnitudes of the vectors: ∣∣ u ∣∣ = 2 and ∣∣ v ∣∣ = 2 .
Use the dot product formula to find the cosine of the angle: cos ( θ ) = 0 .
Determine the angle by taking the inverse cosine: θ = 9 0 ∘ .
Explanation
Problem Analysis We are given two vectors u = ⟨ 1 , − 1 ⟩ and v = ⟨ 1 , 1 ⟩ . Our goal is to find the angle between these two vectors.
Dot Product Formula We can use the dot product formula to find the angle θ between two vectors u and v :
u ⋅ v = ∣∣ u ∣∣ ⋅ ∣∣ v ∣∣ ⋅ cos ( θ )
Calculate Dot Product First, calculate the dot product of u and v :
u ⋅ v = ( 1 ) ( 1 ) + ( − 1 ) ( 1 ) = 1 − 1 = 0
Calculate Magnitude of u Next, calculate the magnitude (or length) of vector u :
∣∣ u ∣∣ = 1 2 + ( − 1 ) 2 = 1 + 1 = 2
Calculate Magnitude of v Then, calculate the magnitude of vector v :
∣∣ v ∣∣ = 1 2 + 1 2 = 1 + 1 = 2
Substitute Values Now, substitute these values into the dot product formula:
0 = ( 2 ) ( 2 ) cos ( θ )
0 = 2 cos ( θ )
Solve for Cosine Solve for cos ( θ ) :
cos ( θ ) = 2 0 = 0
Find the Angle Finally, find the angle θ by taking the inverse cosine (also known as arccos) of 0:
θ = arccos ( 0 ) = 9 0 ∘
Examples
Understanding the angle between vectors is crucial in many fields. For example, in physics, when analyzing forces acting on an object, knowing the angles between the force vectors helps determine the net force. In computer graphics, calculating the angle between light vectors and surface normal vectors is essential for rendering realistic lighting effects. In navigation, the angle between the direction vector of a ship and the wind vector helps in optimizing the sail settings.
