Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈5, 7〉
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8. Vectors
Geometric Vectors
Problem 43b
Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = \langle -1, 2 \rangle \) and \( \mathbf{v} = \langle 3, 0 \rangle \).
Multiply vector \( \mathbf{u} \) by the scalar 2: calculate \( 2\mathbf{u} = 2 \times \langle -1, 2 \rangle \).
Multiply vector \( \mathbf{v} \) by the scalar 3: calculate \( 3\mathbf{v} = 3 \times \langle 3, 0 \rangle \).
Add the resulting vectors from steps 2 and 3 component-wise: \( 2\mathbf{u} + 3\mathbf{v} = \langle 2u_1 + 3v_1, 2u_2 + 3v_2 \rangle \), where \( u_1, u_2 \) and \( v_1, v_2 \) are the components of \( \mathbf{u} \) and \( \mathbf{v} \) respectively.
Write the final vector as the sum of the components found in step 4, which represents \( 2\mathbf{u} + 3\mathbf{v} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition
Vector addition involves combining two vectors by adding their corresponding components. For vectors u = 〈u₁, u₂〉 and v = 〈v₁, v₂〉, the sum u + v is 〈u₁ + v₁, u₂ + v₂〉. This operation results in a new vector representing the combined effect of both.
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Scalar Multiplication of Vectors
Scalar multiplication scales a vector by multiplying each of its components by a real number (scalar). For a scalar k and vector u = 〈u₁, u₂〉, the product k*u is 〈k*u₁, k*u₂〉. This changes the vector's magnitude without altering its direction unless the scalar is negative.
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Component-wise Operations in 2D Vectors
Operations on 2D vectors are performed component-wise, meaning each x and y component is handled separately. This approach simplifies calculations like 2u + 3v by first scaling each vector and then adding corresponding components to find the resultant vector.
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