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

Dot Product

Trigonometry

8. Vectors

Dot Product

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1:51 minutes

Problem 49

Textbook Question

In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither.v = 3i - 5j, w = 6i + 18 j 5

Verified step by step guidance

Step 1: Understand the problem by identifying the vectors v and w. Here, v = 3i - 5j and w = 6i + 18j.

Step 2: Recall that two vectors are parallel if one is a scalar multiple of the other. Check if there exists a scalar k such that v = k * w.

Step 3: Recall that two vectors are orthogonal if their dot product is zero. Calculate the dot product of v and w using the formula: v • w = (3)(6) + (-5)(18).

Step 4: Evaluate the conditions for parallelism and orthogonality based on the results from Steps 2 and 3.

Step 5: Conclude whether the vectors are parallel, orthogonal, or neither based on the evaluations from the previous steps.

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Key Concepts

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

Vector Operations

Understanding vector operations is essential for analyzing the relationship between vectors v and w. This includes addition, subtraction, and scalar multiplication, which help in determining how vectors interact in a coordinate system. In this case, we will focus on the dot product and the concept of direction to assess their relationship.

Dot Product

The dot product of two vectors is a crucial tool for determining their relationship. It is calculated by multiplying corresponding components of the vectors and summing the results. If the dot product is zero, the vectors are orthogonal (perpendicular). If the dot product is positive or negative, it indicates the angle between them, helping to determine if they are parallel or neither.

Parallel and Orthogonal Vectors

Vectors are parallel if they point in the same or opposite directions, which can be determined by checking if one vector is a scalar multiple of the other. Orthogonal vectors, on the other hand, are at right angles to each other, which is confirmed by a dot product of zero. Understanding these definitions is key to classifying the relationship between the given vectors v and w.