Table of contents

0. Review of College Algebra4h 43m
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

Trigonometry

8. Vectors

Geometric Vectors

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2:29 minutes

Problem 31

Textbook Question

In Exercises 21–38, letu = 2i - 5j, v = -3i + 7j, and w = -i - 6j.Find each specified vector or scalar.3w + 2v

Verified step by step guidance

Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \), \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \), and \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).

Determine the expression to solve: \( 3\mathbf{w} + 2\mathbf{v} \).

Calculate \( 3\mathbf{w} \) by multiplying each component of \( \mathbf{w} \) by 3: \( 3(-\mathbf{i} - 6\mathbf{j}) = -3\mathbf{i} - 18\mathbf{j} \).

Calculate \( 2\mathbf{v} \) by multiplying each component of \( \mathbf{v} \) by 2: \( 2(-3\mathbf{i} + 7\mathbf{j}) = -6\mathbf{i} + 14\mathbf{j} \).

Add the results of \( 3\mathbf{w} \) and \( 2\mathbf{v} \) by combining like terms: \((-3\mathbf{i} - 18\mathbf{j}) + (-6\mathbf{i} + 14\mathbf{j})\).

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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 or more vectors to produce a resultant vector. This is done by adding their corresponding components. For example, if vector u = ai + bj and vector v = ci + dj, then u + v = (a+c)i + (b+d)j. Understanding this concept is crucial for solving problems that require the manipulation of multiple vectors.

Scalar Multiplication

Scalar multiplication refers to the process of multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, if k is a scalar and v = ai + bj, then k*v = (ka)i + (kb)j. This concept is essential for adjusting the size of vectors in vector operations, such as in the given problem.

Component Form of Vectors

Vectors can be expressed in component form, which breaks them down into their horizontal and vertical components. For example, a vector v = ai + bj has a horizontal component 'a' and a vertical component 'b'. This representation is vital for performing operations like addition and scalar multiplication, as it allows for straightforward calculations with the individual components.