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Chapter 8: Problem 25

Find a unit vector that is orthogonal to both \([1,1,0]\) and \([1,0,1] .\)

Short Answer

Expert verified
The unit vector is \( \left[ \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right] \).

Step by step solution

01

Calculate Cross Product

To find a vector orthogonal to both given vectors, we can calculate their cross product. Let the vectors be \( \mathbf{a} = [1,1,0] \) and \( \mathbf{b} = [1,0,1] \). The cross product \( \mathbf{a} \times \mathbf{b} \) is computed as follows: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 1 & 0 \ 1 & 0 & 1 \end{vmatrix} = \mathbf{i}(1 \cdot 1 - 0 \cdot 0) - \mathbf{j}(1 \cdot 1 - 0 \cdot 1) + \mathbf{k}(1 \cdot 0 - 1 \cdot 1) \] Thus, the cross product is: \[ \mathbf{a} \times \mathbf{b} = [1, -1, -1] \].
02

Calculate Magnitude of the Cross Product

The magnitude of the vector \( [1, -1, -1] \) can be calculated using the formula \[ ||\mathbf{v}|| = \sqrt{x^2 + y^2 + z^2} \].Substituting the values: \[ ||\mathbf{v}|| = \sqrt{1^2 + (-1)^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \].
03

Normalize the Cross Product

To convert the vector \( [1, -1, -1] \) into a unit vector, divide each component by its magnitude \( \sqrt{3} \).The unit vector is: \[ \left[ \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right] \].

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Key Concepts

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

Cross Product
The cross product is a fundamental operation in vector calculus, especially useful in physics and engineering. It helps us find a vector that is perpendicular to two given vectors in three-dimensional space. When we compute the cross product of two vectors, we use the determinant of a matrix filled with unit vectors and the components of the given vectors.
  • First row: contains unit vectors \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \).
  • Second row: the components of the first vector \( \mathbf{a} \).
  • Third row: the components of the second vector \( \mathbf{b} \).
The determinant calculation will give us a new vector that is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b} \).This vector can provide insight into the orientation and spatial relationship of material objects in life sciences.
Unit Vector
A unit vector is a vector with a magnitude of one. It indicates direction but doesn't have any unit of length. Transforming any given vector into a unit vector is known as normalization.
An important application of unit vectors is in defining directions regardless of scale, which is crucial in fields such as biology and ecology.
  • First, calculate the magnitude of the vector \( \mathbf{v} \) using: \( ||\mathbf{v}|| = \sqrt{x^2 + y^2 + z^2} \).
  • Then, divide each component of the vector by this magnitude. This results in scaling the entire vector to have a length of one while maintaining its direction.
In our example, the cross product \([1, -1, -1]\) is converted to \( \left[ \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right] \), a unit vector.
Orthogonal Vectors
Orthogonal vectors are vectors that meet at right angles. In mathematical terms, two vectors are orthogonal if their dot product is zero.
For vectors in three-dimensional space, being orthogonal means that they have no components in common along any of the axes, similar to how coordinate axes intersect perpendicularly.
  • Calculating a cross product is a handy tool to find vectors that are orthogonal, as the resultant vector from a cross product is perpendicular to the two original vectors.
  • Orthogonality is a fundamental concept in various life sciences applications such as population genetics and clinical trials, as it helps in understanding the independence of traits or effects.
Understanding and applying the concept of orthogonality ensures better modeling and analysis for researchers across biological sciences.

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