Given vectors u = ⟨2, –3⟩ and v = ⟨1, –1⟩, what is the measure of the angle between the vectors?
A. 11.3°
B. 78.9°
C. 92.2°
D. 101.3°
Answer (2)
To find the measure of the angle between the vectors u = ⟨ 2 , − 3 ⟩ and v = ⟨ 1 , − 1 ⟩ , we can use the dot product formula along with the cosine of the angle. The formula is:
cos ( θ ) = ∥ u ∥∥ v ∥ u ⋅ v
Step 1: Calculate the Dot Product, u ⋅ v :
u ⋅ v = ( 2 ) ( 1 ) + ( − 3 ) ( − 1 ) = 2 + 3 = 5
Step 2: Calculate the Magnitude of u , ∥ u ∥ :
∥ u ∥ = 2 2 + ( − 3 ) 2 = 4 + 9 = 13
Step 3: Calculate the Magnitude of v , ∥ v ∥ :
∥ v ∥ = 1 2 + ( − 1 ) 2 = 1 + 1 = 2
Step 4: Find cos ( θ ) :
cos ( θ ) = 13 × 2 5 = 26 5
Step 5: Find the Angle θ :
To find θ , take the arccosine (inverse cosine) of the result:
θ = cos − 1 ( 26 5 )
Calculating this gives an approximate measure of θ to be:
θ ≈ 78. 9 ∘
Therefore, the measure of the angle between the vectors u and v is approximately 78. 9 ∘ . The correct answer is (B) 78.9°.
The measure of the angle between the vectors u = ⟨ 2 , − 3 ⟩ and v = ⟨ 1 , − 1 ⟩ is approximately 78. 9 ∘ . Therefore, the correct answer is (B) 78.9°.
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