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Magazine Chapter 3: Problem 19
Find \(\left\|p r o j_{a} u\right\|\) (a) \(\mathbf{u}=(1,-2), \mathbf{a}=(-4,-3)\) (b) \(\mathbf{u}=(3,0,4), \mathbf{a}=(2,3,3)\)
Short Answer
Expert verified
(a) 0.4; (b) \( \approx 4.486 \)
Step by step solution
01
Understand the Projection Formula
The projection of vector \( \mathbf{u} \) onto vector \( \mathbf{a} \) is given by the formula: \( \text{proj}_{\mathbf{a}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \cdot \mathbf{a} \). The magnitude of the projection is \( \left\| \text{proj}_{\mathbf{a}} \mathbf{u} \right\| \).
02
Calculate Dot Products for (a)
For (a), find \( \mathbf{u} \cdot \mathbf{a} = (1)(-4) + (-2)(-3) = -4 + 6 = 2 \). \( \mathbf{a} \cdot \mathbf{a} = (-4)^2 + (-3)^2 = 16 + 9 = 25 \).
03
Compute Projection Magnitude for (a)
Use the values from Step 2 in the formula: \( \left\| \text{proj}_{\mathbf{a}} \mathbf{u} \right\| = \left| \frac{\mathbf{u} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \right| \cdot \left\| \mathbf{a} \right\| = \left| \frac{2}{25} \right| \cdot \sqrt{25} = \left| \frac{2}{25} \right| \cdot 5 = \frac{10}{25} = 0.4 \).
04
Calculate Dot Products for (b)
For (b), find \( \mathbf{u} \cdot \mathbf{a} = (3)(2) + (0)(3) + (4)(3) = 6 + 0 + 12 = 18 \). \( \mathbf{a} \cdot \mathbf{a} = (2)^2 + (3)^2 + (3)^2 = 4 + 9 + 9 = 22 \).
05
Compute Projection Magnitude for (b)
Use the values from Step 4 in the formula: \( \left\| \text{proj}_{\mathbf{a}} \mathbf{u} \right\| = \left| \frac{18}{22} \right| \cdot \left\| \mathbf{a} \right\| = \left| \frac{18}{22} \right| \cdot \sqrt{22} \). First, simplify the fraction to \( \frac{9}{11} \). So, \( \left\| \mathbf{a} \right\| = \sqrt{22} \) and the magnitude is \( \frac{9}{11} \cdot \sqrt{22} \approx 4.486 \).
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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 fundamental operation in linear algebra involving two vectors. Given two vectors, \( \mathbf{u} = (u_1, u_2, \ldots, u_n) \) and \( \mathbf{a} = (a_1, a_2, \ldots, a_n) \), the dot product \( \mathbf{u} \cdot \mathbf{a} \) is calculated as the sum of the products of their corresponding components:\[ \mathbf{u} \cdot \mathbf{a} = u_1a_1 + u_2a_2 + \cdots + u_na_n \]The dot product results in a single number (a scalar), which gives a measure of how much one vector goes in the direction of the other.
It has a number of useful properties:
It has a number of useful properties:
- If the dot product is zero, the vectors are perpendicular.
- The dot product can also determine the angle between two vectors through the formula: \( \mathbf{u} \cdot \mathbf{a} = \|\mathbf{u}\| \|\mathbf{a}\| \cos\theta \), where \( \theta \) is the angle between the vectors.
- It is commutative, meaning \( \mathbf{u} \cdot \mathbf{a} = \mathbf{a} \cdot \mathbf{u} \).
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It provides the framework for solving a system of linear equations, understanding multi-dimensional spaces, and performing vector operations like dot and cross products.
Key elements include:
Key elements include:
- Vectors: Objects represented by an order of numbers, which can be added, subtracted, and scaled.
- Matrices: Rectangular arrays of numbers used to represent linear transformations and solve linear systems.
- Applications: Linear algebra is essential in fields like computer science, physics, engineering, and economics. Movements in space, representation of systems, and transformations are just a few areas that benefit greatly from its concepts.
Vector Magnitude
The magnitude of a vector, sometimes called the length or norm, is a measure of how long a vector is. For a vector \( \mathbf{a} = (a_1, a_2, \ldots, a_n) \), its magnitude \( \| \mathbf{a} \| \) is calculated using the formula:\[ \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2 + \cdots + a_n^2} \]It is always a non-negative value, reflecting the 'size' of the vector in space.
Vector magnitude is important because:
Vector magnitude is important because:
- It provides a scale for the vector, making it possible to compare with other vectors.
- In physics, it represents quantities like force, velocity, and displacement.
- In projection, it's used to find how much of one vector projects onto another, necessary in operations like calculating the magnitude of vector projections.
