Select the correct answer. Vector $u$ has a magnitude of 5 units and a direction angle of $75^{\circ}$, and vector $v$ has a magnitude of 6 units and a direction angle of $210^{\circ}$. What is the component form of the vector $u + v$ ? A. $\langle-3.95,1.83\rangle$ B. $\langle-3.94,1.89\rangle$ C. $\langle-3.90,1.89\rangle$ D. $\langle-3.90,1.83\rangle$
Answer (2)
Find the component form of vector u : u = ⟨ 5 cos ( 7 5 ∘ ) , 5 sin ( 7 5 ∘ )⟩ .
Find the component form of vector v : v = ⟨ 6 cos ( 21 0 ∘ ) , 6 sin ( 21 0 ∘ )⟩ .
Add the component forms: u + v = ⟨ 5 cos ( 7 5 ∘ ) + 6 cos ( 21 0 ∘ ) , 5 sin ( 7 5 ∘ ) + 6 sin ( 21 0 ∘ )⟩ .
Calculate the result: ⟨ − 3.90 , 1.83 ⟩ .
Explanation
Problem Analysis We are given two vectors, u and v . Vector u has a magnitude of 5 and a direction angle of 7 5 ∘ , and vector v has a magnitude of 6 and a direction angle of 21 0 ∘ . We want to find the component form of the vector u + v .
Finding the component form of vector u First, we need to find the component form of vector u . The x -component of u is given by 5 cos ( 7 5 ∘ ) and the y -component is given by 5 sin ( 7 5 ∘ ) . So, u = ⟨ 5 cos ( 7 5 ∘ ) , 5 sin ( 7 5 ∘ )⟩ .
Finding the component form of vector v Next, we need to find the component form of vector v . The x -component of v is given by 6 cos ( 21 0 ∘ ) and the y -component is given by 6 sin ( 21 0 ∘ ) . So, v = ⟨ 6 cos ( 21 0 ∘ ) , 6 sin ( 21 0 ∘ )⟩ .
Adding the component forms of u and v Now, we add the component forms of vectors u and v to find the component form of u + v . This is given by u + v = ⟨ 5 cos ( 7 5 ∘ ) + 6 cos ( 21 0 ∘ ) , 5 sin ( 7 5 ∘ ) + 6 sin ( 21 0 ∘ )⟩ .
Calculating the values We calculate the values: 5 cos ( 7 5 ∘ ) ≈ 1.294 5 sin ( 7 5 ∘ ) ≈ 4.830 6 cos ( 21 0 ∘ ) ≈ − 5.196 6 sin ( 21 0 ∘ ) = − 3 Therefore, u + v ≈ ⟨ 1.294 − 5.196 , 4.830 − 3 ⟩ = ⟨ − 3.902 , 1.830 ⟩ .
Selecting the correct answer Comparing the result with the given options, we see that the closest answer is ⟨ − 3.90 , 1.83 ⟩ .
Examples
Vector addition is used extensively in physics to calculate resultant forces, velocities, and accelerations. For example, if you have two forces acting on an object, you can represent them as vectors and add them to find the net force. This is crucial in understanding the motion of objects under multiple influences, such as a boat being propelled by its engine while being pushed by the wind and pulled by the water current. By adding these vectors, we can determine the boat's overall direction and speed.
To find the component form of the sum of vectors u and v , we first calculate their individual components based on their magnitudes and angles. Adding these components gives us approximately egin{pmatrix} -3.90, 1.83 \\} . Therefore, the correct answer is option D: \langle -3.90, 1.83 \rangle.
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