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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1:40 minutes

Problem 23

Textbook Question

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.v = i + j, w = i - j

Verified step by step guidance

Step 1: Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if their dot product is zero.

Step 2: Write the vectors \( \mathbf{v} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} - \mathbf{j} \) in component form: \( \mathbf{v} = (1, 1) \) and \( \mathbf{w} = (1, -1) \).

Step 3: Use the formula for the dot product of two vectors \( \mathbf{v} = (a_1, b_1) \) and \( \mathbf{w} = (a_2, b_2) \), which is \( \mathbf{v} \cdot \mathbf{w} = a_1a_2 + b_1b_2 \).

Step 4: Substitute the components of \( \mathbf{v} \) and \( \mathbf{w} \) into the dot product formula: \( (1)(1) + (1)(-1) \).

Step 5: Simplify the expression to determine if the dot product is zero, which will confirm if the vectors are orthogonal.

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Key Concepts

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

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. For vectors v = (v1, v2) and w = (w1, w2), the dot product is given by v · w = v1*w1 + v2*w2. This operation is crucial for determining the angle between vectors and checking for orthogonality.

Orthogonal Vectors

Two vectors are considered orthogonal if they are perpendicular to each other, which occurs when their dot product equals zero. This property is significant in various applications, including physics and computer graphics, as it indicates that the vectors do not influence each other in their respective directions. Understanding orthogonality helps in simplifying problems involving vector components.

Vector Representation

Vectors can be represented in terms of their components along the coordinate axes. In this case, the vectors v = i + j and w = i - j can be expressed as v = (1, 1) and w = (1, -1). This representation allows for straightforward calculations, such as the dot product, and aids in visualizing the vectors in a Cartesian plane, facilitating the analysis of their relationships.