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Magazine Chapter 3: Problem 75
Find (a) "north cross west," (b) "down dot south," (c) "east cross up," (d) "west dot west," and (e) "south cross south." Let each "vector" have unit magnitude.
Short Answer
Expert verified
(a) -\( \hat{k} \), (b) 0, (c) -\( \hat{j} \), (d) 1, (e) \( \vec{0} \).
Step by step solution
01
Understanding Directional Vectors
In this problem, we're dealing with directional vectors where each direction (north, south, east, west, up, down) has a unit magnitude. This means that each vector is represented as a unit vector on a coordinate system. For convenience, we can represent north as \( \hat{i} \), east as \( \hat{j} \), up as \( \hat{k} \), and so on, where each of these vectors has a length of 1.
02
Calculate "north cross west"
The cross product of two vectors gives a vector that is perpendicular to both. If we specify north as \( \hat{j} \) and west as \( -\hat{i} \), then \( \hat{j} \times -\hat{i} = -\hat{k} \). Therefore, "north cross west" results in the vector \( -\hat{k} \).
03
Calculate "down dot south"
The dot product of two perpendicular unit vectors is 0. So, if down is -\( \hat{k} \) and south is -\( \hat{j} \), then \( (-\hat{k}) \cdot (-\hat{j}) = 0 \). Hence, "down dot south" is 0.
04
Calculate "east cross up"
For this cross product, we use east as \( \hat{i} \) and up as \( \hat{k} \). The result, \( \hat{i} \times \hat{k} = -\hat{j} \), indicates "east cross up" is \( -\hat{j} \).
05
Calculate "west dot west"
The dot product of a unit vector with itself is 1. If west is -\( \hat{i} \), then \( (-\hat{i}) \cdot (-\hat{i}) = 1 \). Therefore, "west dot west" is 1.
06
Calculate "south cross south"
The cross product of any vector with itself is always the zero vector. If south is -\( \hat{j} \), then \( (-\hat{j}) \times (-\hat{j}) = \vec{0} \). Hence, "south cross south" is \( \vec{0} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Directional Vectors
Directional vectors are fundamental in understanding the movements in 3D space. When you think about directions like north, south, east, west, up, and down, you can imagine these as arrows pointing in specific directions on a coordinate system. Each of these directions can be represented by unit vectors, which have a length of one.
- North: Typically represented as \( \hat{j} \).
- East: Represented as \( \hat{i} \).
- Up: Shown as \( \hat{k} \).
- South: Opposite of north, as \( -\hat{j} \).
- West: Opposite of east, as \( -\hat{i} \).
- Down: Opposite of up, as \( -\hat{k} \).
Cross Product
The cross product is a powerful tool in vector operations. When you take the cross product of two vectors, the result is another vector that is perpendicular to the plane formed by the original two vectors. This operation is very useful in physics and engineering to find vectors orthogonal to given planes.
When calculating the cross product:
When calculating the cross product:
- The order matters: \( \vec{A} \times \vec{B} eq \vec{B} \times \vec{A} \).
- The result is a vector: For vectors \( \hat{j} \) (north) and \( -\hat{i} \) (west), their cross product is \( -\hat{k} \). This vector points in the direction perpendicular to both north and west.
- If the same direction vectors are used, like \( \vec{C} \times \vec{C} \), the result is the zero vector \( \vec{0} \), as there鈥檚 no defined plane.
Dot Product
The dot product is one of the most straightforward vector operations. It allows you to measure how much one vector extends in the direction of another. The dot product turns two vectors into a scalar (a single number), which can be particularly useful for calculating work done or projection calculations.
To compute the dot product:
To compute the dot product:
- The product is zero if vectors are perpendicular, as they have no common direction. For instance, \( (-\hat{k}) \cdot (-\hat{j}) = 0 \).
- It's equal to the product of the magnitudes of the vectors times the cosine of the angle between them. For vectors in the same direction, the dot product is the square of their magnitude. For \( -\hat{i} \), the dot product with itself is \( 1 \).
Unit Vectors
Unit vectors are the building blocks of vector mathematics. Having a length of one, unit vectors provide the direction without affecting the magnitude of calculations. They serve as the default vectors along the coordinate axes, playing a crucial role in vector normalization.
- A unit vector in any direction maintains its magnitude as one.
- They simplify vector expressions and calculations. When you represent complex vectors using unit vectors, it allows for easier operations such as addition and scalar multiplication.
- In 3D space, the primary unit vectors include \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \).
