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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1:44 minutes

Problem 27

Textbook Question

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

Verified step by step guidance

Identify the given vector \( v = -3\mathbf{i} + 7\mathbf{j} \).

Understand that the problem asks for the scalar multiplication of vector \( v \) by 5.

Apply the scalar multiplication: Multiply each component of vector \( v \) by 5.

Calculate the new \( i \)-component: \( 5 \times (-3) \).

Calculate the new \( j \)-component: \( 5 \times 7 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Operations

Vector operations involve mathematical manipulations of vectors, such as addition, subtraction, and scalar multiplication. In this case, multiplying vector v by a scalar (5) means scaling its components by that number, which affects both the magnitude and direction of the vector.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a real number). This operation results in a new vector whose direction remains the same if the scalar is positive, but its magnitude is scaled by the absolute value of the scalar. For example, multiplying vector v by 5 will increase its length while maintaining its direction.

Component Form of Vectors

Vectors can be expressed in component form, typically as a combination of unit vectors i and j in a two-dimensional space. For instance, vector v = -3i + 7j indicates that it has a horizontal component of -3 and a vertical component of 7. Understanding this representation is crucial for performing operations like scalar multiplication.