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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2:44 minutes

Problem 33

Textbook Question

In Exercises 21–38, letu = 2i - 5j, v = -3i + 7j, and w = -i - 6j.Find each specified vector or scalar.3v - 4w

Verified step by step guidance

Identify the given vectors: \( v = -3i + 7j \) and \( w = -i - 6j \).

Multiply vector \( v \) by 3: \( 3v = 3(-3i + 7j) = -9i + 21j \).

Multiply vector \( w \) by 4: \( 4w = 4(-i - 6j) = -4i - 24j \).

Subtract \( 4w \) from \( 3v \): \( 3v - 4w = (-9i + 21j) - (-4i - 24j) \).

Combine like terms: \( (-9i + 4i) + (21j + 24j) \).

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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 combining two or more vectors by adding their corresponding components. Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. Understanding these operations is essential for manipulating vectors in problems like the one presented.

Component Form of Vectors

Vectors can be expressed in component form, typically as 'ai + bj', where 'a' and 'b' are the horizontal and vertical components, respectively. This representation allows for straightforward calculations, such as addition and scalar multiplication, by simply performing arithmetic on the components. In this question, the vectors u, v, and w are given in this form.

Resultant Vector

The resultant vector is the vector that results from the addition of two or more vectors. It represents the cumulative effect of the individual vectors. In the context of the question, calculating 3v - 4w involves finding the resultant vector after scaling vectors v and w and then performing vector subtraction.