Domanda del libro di testo
In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = -6i - 5j, w = -10i - 8j
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 4, Problema 5
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w
Guida verificata passo dopo passo1
Identify the given vectors: \( \mathbf{v} = -5\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{w} = 2\mathbf{i} - 4\mathbf{j} \).
Calculate the scalar multiplication of vector \( \mathbf{v} \) by 3: multiply each component of \( \mathbf{v} \) by 3, resulting in \( 3\mathbf{v} = 3(-5\mathbf{i}) + 3(2\mathbf{j}) \).
Calculate the scalar multiplication of vector \( \mathbf{w} \) by 4: multiply each component of \( \mathbf{w} \) by 4, resulting in \( 4\mathbf{w} = 4(2\mathbf{i}) + 4(-4\mathbf{j}) \).
Form the expression \( 3\mathbf{v} - 4\mathbf{w} \) by subtracting the components of \( 4\mathbf{w} \) from the corresponding components of \( 3\mathbf{v} \).
Combine the resulting components to write the final vector in the form \( a\mathbf{i} + b\mathbf{j} \), where \( a \) and \( b \) are the calculated values.
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Vector Addition and Scalar Multiplication
Vector addition involves adding corresponding components of vectors, while scalar multiplication scales each component by a given number. For example, multiplying vector v by 3 means multiplying each component of v by 3. These operations are fundamental for combining and scaling vectors.
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Percorso guidato
05:05
Multiplying Vectors By Scalars
Component Form of Vectors
Vectors in two dimensions can be expressed in component form as v = ai + bj, where a and b are the components along the x and y axes, respectively. Understanding this form allows for straightforward arithmetic operations on vectors by handling their components separately.
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03:55Position Vectors & Component Form
Linear Combination of Vectors
A linear combination involves multiplying vectors by scalars and then adding the results. In this problem, 3v - 4w is a linear combination, meaning you multiply v by 3, w by 4, and subtract the results component-wise to find the resulting vector.
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7:48Solving Linear Equations
Pratica correlata
Domanda del libro di testo
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
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Domanda del libro di testo
In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β.
c. Find the exact value of the expression.
cos 50° cos 20° + sin 50° sin 20°
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Domanda del libro di testo
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w
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Domanda del libro di testo
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
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Domanda del libro di testo
In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = 3i + j
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