Given vectors $u =\langle-2,3\rangle$ and $v =\langle-6,9)$, what is the measure of the angle between $u$ and $v$?
A. $0^{\circ}$
B. $86^{\circ}$
C. $91^{\circ}$
D. $180^{\circ}$
Answer (1)
∙ Calculate the dot product of vectors u and v : u ⋅ v = 39 .
∙ Determine the magnitudes of u and v : ∣∣ u ∣∣ = 13 and ∣∣ v ∣∣ = 3 13 .
∙ Apply the formula cos ( θ ) = ∣∣ u ∣∣ ⋅ ∣∣ v ∣∣ u ⋅ v to find cos ( θ ) = 1 .
∙ Calculate the angle θ by taking the inverse cosine: θ = arccos ( 1 ) = 0 ∘ .
Explanation
Problem Setup We are given two vectors u = ⟨ − 2 , 3 ⟩ and v = ⟨ − 6 , 9 ⟩ . Our goal is to find the angle between these two vectors.
Calculate the Dot Product First, let's calculate the dot product of the two vectors: u ⋅ v = ( − 2 ) × ( − 6 ) + ( 3 ) × ( 9 ) = 12 + 27 = 39
Calculate the Magnitudes Next, we need to find the magnitudes of the vectors u and v .
∣∣ u ∣∣ = ( − 2 ) 2 + 3 2 = 4 + 9 = 13 ≈ 3.6055 ∣∣ v ∣∣ = ( − 6 ) 2 + 9 2 = 36 + 81 = 117 = 9 × 13 = 3 13 ≈ 10.8167
Apply the Angle Formula Now, we can use the formula for the cosine of the angle θ between two vectors: cos ( θ ) = ∣∣ u ∣∣ ⋅ ∣∣ v ∣∣ u ⋅ v = 13 ⋅ 3 13 39 = 3 × 13 39 = 39 39 = 1 So, cos ( θ ) = 1 .
Find the Angle To find the angle θ , we take the inverse cosine (arccos) of 1: θ = arccos ( 1 ) = 0 radians Converting to degrees, we have: θ = 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 angle between the force vectors helps determine the net force. In computer graphics, calculating the angle between vectors is essential for lighting and shading models, creating realistic visual effects. In navigation, the angle between two displacement vectors can help determine the most efficient path.
