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Magazine Chapter 12: Problem 41
Show that the vector orth \(_{\mathrm{a}} \mathbf{b}=\mathbf{b}-\) proja \(\mathbf{b}\) is orthogonal to a. (It is called an orthogonal projection of b.)
Short Answer
Expert verified
The vector \( \text{orth}_{\mathbf{a}} \mathbf{b} \) is orthogonal to \( \mathbf{a} \) because their dot product is zero.
Step by step solution
01
Define Vector Projection
The projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \) is calculated using the formula: \[ \text{proj}_\mathbf{a} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \] This represents the component of \( \mathbf{b} \) that is parallel to \( \mathbf{a} \).
02
Define Orthogonal Projection
The orthogonal projection of \( \mathbf{b} \) onto \( \mathbf{a} \) is given by: \[ \text{orth}_{\mathbf{a}} \mathbf{b} = \mathbf{b} - \text{proj}_\mathbf{a} \mathbf{b} \] This equation gives the component of \( \mathbf{b} \) that is orthogonal to \( \mathbf{a} \).
03
Substitute the Projection Formula
Substitute the formula for projection into the expression for orthogonal projection: \[ \text{orth}_{\mathbf{a}} \mathbf{b} = \mathbf{b} - \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \right) \] This expands the definition of the orthogonal vector we are examining.
04
Check Orthogonality
To verify that \( \text{orth}_{\mathbf{a}} \mathbf{b} \) is orthogonal to \( \mathbf{a} \), calculate the dot product: \[ \mathbf{a} \cdot (\mathbf{b} - \text{proj}_\mathbf{a} \mathbf{b}) = \mathbf{a} \cdot \mathbf{b} - \mathbf{a} \cdot \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \right) \]
05
Simplify and Conclude
Simplify the expression from Step 4: \[ \mathbf{a} \cdot \mathbf{b} - \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} (\mathbf{a} \cdot \mathbf{a}) = \mathbf{a} \cdot \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) = 0 \] Since the dot product is zero, \( \text{orth}_{\mathbf{a}} \mathbf{b} \) is orthogonal to \( \mathbf{a} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dot Product
The dot product is a crucial operation in vector algebra. It enables us to explore the relationship between two vectors. If we have two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is defined as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 + ... + a_nb_n \). Here, each \( a_i \) and \( b_i \) represent components of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.
- The dot product is a scalar, not a vector. It gives a single number as an output.
- It is fundamental in determining the angle between two vectors. When the dot product is zero, the vectors are orthogonal (i.e., at a 90-degree angle).
Vector Projection
Understanding vector projection helps us to see how one vector lies relative to another. It involves "projecting" a vector onto another vector. For instance, the projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \) is given by:\[\text{proj}_\mathbf{a} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}\]
- This gives a new vector that points in the direction of \( \mathbf{a} \), showing how much of \( \mathbf{b} \) aligns with \( \mathbf{a} \).
- It's useful for determining components that are parallel in geometry, physics, and engineering applications.
Orthogonal Projection
The orthogonal projection of a vector aids in separating it into two distinct components: one parallel and one perpendicular to another vector. For vector \( \mathbf{b} \) onto \( \mathbf{a} \), it is defined as:\[\text{orth}_{\mathbf{a}} \mathbf{b} = \mathbf{b} - \text{proj}_\mathbf{a} \mathbf{b}\]This yields the component of \( \mathbf{b} \) that stands orthogonal to \( \mathbf{a} \).
- Any vector can be split into parallel and orthogonal parts relative to another vector.
- Orthogonal components play a key role in simplifying calculations in physics, such as decomposing forces.
Linear Algebra
Linear Algebra is a branch of mathematics that studies vectors, vector spaces (also known as linear spaces), and linear transformations. It lays the foundation for many concepts in mathematics, including the ones mentioned above.
- Central to linear algebra is the study of linear equations and the matrices representing these equations.
- It deals with operations like addition and scalar multiplication of vectors and matrices, solving systems of linear equations, and many transformations.
