In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.
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- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
8. Vectors
Geometric Vectors
Problem 5
Textbook Question
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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