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

Previous problemNext problem

2:04 minutes

Problem 29

Textbook Question

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.v = 3i, w = -4i

Verified step by step guidance

Identify the vectors \( \mathbf{v} = 3\mathbf{i} \) and \( \mathbf{w} = -4\mathbf{i} \).

Recall that two vectors are orthogonal if their dot product is zero.

The dot product of two vectors \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \) is given by \( \mathbf{v} \cdot \mathbf{w} = ac + bd \).

Substitute the components of \( \mathbf{v} \) and \( \mathbf{w} \) into the dot product formula: \( 3 \times (-4) + 0 \times 0 \).

Evaluate the expression to determine if the dot product is zero, which would indicate that the vectors are orthogonal.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 the corresponding components of the vectors and summing the results. For vectors v = ai + bj and w = ci + dj, the dot product is given by v · w = ac + bd. This operation is crucial for determining the angle between vectors and checking for orthogonality.

Orthogonal Vectors

Two vectors are considered orthogonal if their dot product equals zero. This means that they are at right angles (90 degrees) to each other in a geometric sense. In practical terms, if v · w = 0, then the vectors do not influence each other in their respective directions, which is a key concept in various applications, including physics and engineering.

Vector Representation

Vectors can be represented in a coordinate system using unit vectors, such as i, j, and k for three-dimensional space. In this case, v = 3i and w = -4i are both one-dimensional vectors along the x-axis. Understanding vector representation is essential for performing operations like the dot product and visualizing the relationship between vectors in space.