The magnitude of vector $u$ is 34, and the magnitude of vector $v$ is 20. The dot product $u \cdot v =-10$. What is the approximate angle between the vectors?
A. $79^{\circ}$
B. $89^{\circ}$
C. $91^{\circ}$
D. $101^{\circ}$
Answer (1)
Use the dot product formula to relate the magnitudes of the vectors and the angle between them.
Substitute the given values into the dot product formula and solve for cos ( θ ) .
Calculate the angle θ by taking the inverse cosine of the result.
Approximate the value of θ in degrees and choose the closest option: 9 1 ∘ .
Explanation
Problem Analysis We are given the magnitudes of two vectors, ∣∣ u ∣∣ = 34 and ∣∣ v ∣∣ = 20 , and their dot product u ⋅ v = − 10 . We want to find the angle θ between these vectors.
Dot Product Formula The dot product of two vectors is related to their magnitudes and the angle between them by the formula: u ⋅ v = ∣∣ u ∣∣ ⋅ ∣∣ v ∣∣ ⋅ cos ( θ ) We can rearrange this formula to solve for cos ( θ ) :
cos ( θ ) = ∣∣ u ∣∣ ⋅ ∣∣ v ∣∣ u ⋅ v
Substitute Values Substitute the given values into the formula: cos ( θ ) = 34 ⋅ 20 − 10 = 680 − 10 = − 68 1 Now, we find the angle θ by taking the inverse cosine (arccos) of − 68 1 :
θ = arccos ( − 68 1 )
Calculate Angle Using a calculator, we find the approximate value of θ in degrees: θ ≈ 90.8 4 ∘ Comparing this value with the given options, the closest approximation is 9 1 ∘ .
Examples
Understanding the angle between vectors is crucial in physics, especially when analyzing forces. For instance, imagine two people pushing a box. If one person pushes with a force of 34 Newtons and the other with 20 Newtons, and the combined effect results in a dot product of -10 (indicating they're working slightly against each other), calculating the angle helps determine the efficiency of their combined effort. This principle extends to various scenarios, such as optimizing the thrust angle of rocket engines or analyzing magnetic forces.
