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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblema 43b
Capitolo 8, Problema 43b

Given vectors u and v, find: 2u + 3v. 
u = 〈-1, 2〉, v = 〈3, 0〉

Guida verificata passo dopo passo
1
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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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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Adding Vectors Geometrically

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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Multiplying Vectors By Scalars

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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Algebraic Operations on Vectors
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