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Chapter 2: Problem 23

If the dot product of two vectors vanishes, what can you say about their directions?

Short Answer

Expert verified
If the dot product of two vectors is zero, it means that the vectors are perpendicular or orthogonal to each other.

Step by step solution

01

Understanding Dot Product

The dot product, also known as scalar product of two vectors is basically the product of magnitudes of two vectors and the cosine of the angle between them. It is represented as A.B = |A||B|cos胃, where |A| and |B| represent the magnitudes of vectors A and B respectively, and 胃 is the angle between them.
02

Given Condition

According to the condition in the exercise, the dot product of two vectors is zero. This implies that A.B = 0.
03

Inferring the direction of vectors

If the dot product of two vectors is zero, using the formula for dot product, we can infer that A.B = 0 = |A||B|cos胃. This can only happen when cos胃 = 0. This is true if and only if the angle (胃) between the vectors is either 90 or 270 degrees, i.e., vectors are perpendicular. Therefore, the vectors are orthogonal or perpendicular to each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dot Product
When we talk about the dot product of two vectors, we are referring to an operation that results in a single number, or scalar. This is why it's sometimes called the scalar product.
  • The formula for the dot product is given by: \( A \cdot B = |A| |B| \cos(\theta) \), where \(|A|\) and \(|B|\) are the magnitudes of vectors \(A\) and \(B\), and \(\theta\) is the angle between them.
  • If the cosine of \(\theta\) ends up being zero, the dot product itself becomes zero.
Understanding this concept is crucial, because the dot product helps in determining the relationship between two vectors. If the dot product is zero, it tells us that the vectors are orthogonal, which means they are perpendicular to each other. This property is utilized frequently in geometry and physics to verify perpendicularity in problems involving vector components.
Vector Magnitude
To comprehend vectors' magnitudes, visualize them as arrows in space. The magnitude, or length, of a vector gives us an idea of how far this arrow stretches from its starting point.
  • This concept can be calculated using the formula: \( |A| = \sqrt{a_1^2 + a_2^2 + a_3^2} \) for a vector \( A = (a_1, a_2, a_3) \) in a 3-dimensional space.
  • Magnitude is a non-negative value because it represents a length and thus cannot be negative.
Knowing the magnitudes of vectors is essential when working with the dot product, as both vectors' magnitudes directly factor into the calculation. Whether in pure mathematics or applied fields like engineering, vector magnitudes highlight how influential a vector is in its direction.
Angle Between Vectors
The angle between vectors is a vital part in understanding vector interactions. It is closely linked to how vectors add together or intersect each other.
  • The angle \(\theta\) between two vectors can be derived from the dot product formula: \( \cos(\theta) = \frac{A \cdot B}{|A| |B|} \).
  • If \(\cos(\theta) = 0\), then the angle \(\theta\) is either 90 degrees or 270 degrees, indicating that the vectors are perpendicular.
Perpendicular vectors play a fundamental role as they describe a situation where the vectors have zero influence on each other in the direction of their respective magnitudes. This can be visualized easily on a Cartesian plane, where perpendicular vectors form an "L" shape, symbolizing orthogonality.

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Most popular questions from this chapter

If the polar coordinates of a point are \((r, \varphi)\) and its rectangular coordinates are \((x, y),\) determine the polar coordinates of the following points: (a) \((-x, y),\) (b) \((-2 x,\) \(-2 y\) ), and \((c)(3 x,-3 y)\)

Can the magnitude of a particle's displacement be greater that the distance traveled?

A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point \(b(4.0 \mathrm{m}, 1.5 \mathrm{m}, 2.5 \mathrm{m})\) to point \(e(1.0\) m, 4.5 m, 0.5 m). Find the scalar components of the fly's displacement vector and express its displacement vector in vector component form. What is its magnitude?

A delivery man starts at the post office, drives 40 km north, then \(20 \mathrm{km}\) west, then \(60 \mathrm{km}\) northeast, and finally \(50 \mathrm{km}\) north to stop for lunch. Use a graphical method to find his net displacement vector.

Can a magnitude of a vector be negative?
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