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Chapter 10: Problem 24

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-2 \mathbf{j} $$

Short Answer

Expert verified
The parallel vector \(\mathbf{v}_{1}\) is \(\frac{4}{5} \mathbf{i} - \frac{8}{5} \mathbf{j}\) and the orthogonal vector \(\mathbf{v}_{2}\) is \(\frac{6}{5} \mathbf{i} + \frac{3}{5} \mathbf{j}\).

Step by step solution

01

- Understanding Vectors

Given vectors are \(\mathbf{v} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{w} = \mathbf{i} - 2\mathbf{j}\). We need to decompose \(\mathbf{v}\) into a parallel component \(\mathbf{v}_{1}\) and an orthogonal component \(\mathbf{v}_{2}\) with respect to \(\mathbf{w}\).
02

- Find \(\mathbf{v}_{1}\)

The vector \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and can be written as \(k\mathbf{w}\) for some scalar \(k\). To find \(k\), use the formula for projection: \(\mathbf{v}_{1} = \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}\). Calculate the dot products \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{w} \cdot \mathbf{w}\).
03

Step 2.1 - Compute Dot Products

Calculate the dot product \(\mathbf{v} \cdot \mathbf{w}\):\( \mathbf{v} \cdot \mathbf{w} = (2\mathbf{i} - \mathbf{j}) \cdot (\mathbf{i} - 2\mathbf{j}) = 2 \cdot 1 + (-1) \cdot (-2) = 2 + 2 = 4\).Next, calculate \(\mathbf{w} \cdot \mathbf{w}\):\( \mathbf{w} \cdot \mathbf{w} = (\mathbf{i} - 2\mathbf{j}) \cdot (\mathbf{i} - 2\mathbf{j}) = 1 + 4 = 5\).
04

- Calculate \(\mathbf{v}_{1}\)

Using the values from Step 2.1, \(\mathbf{v}_{1} = \frac{4}{5} \mathbf{w} = \frac{4}{5} (\mathbf{i} - 2\mathbf{j}) = \frac{4}{5} \mathbf{i} - \frac{8}{5} \mathbf{j}\).
05

- Find \(\mathbf{v}_{2}\)

By definition, \(\mathbf{v} = \mathbf{v}_{1} + \mathbf{v}_{2}\). Therefore, \(\mathbf{v}_{2} = \mathbf{v} - \mathbf{v}_{1}\). Substitute the values of \(\mathbf{v}\) and \(\mathbf{v}_{1}\): \(\mathbf{v}_{2} = (2\mathbf{i} - \mathbf{j}) - (\frac{4}{5} \mathbf{i} - \frac{8}{5} \mathbf{j}) = (2 - \frac{4}{5})\mathbf{i} - (1 - \frac{8}{5})\mathbf{j}\). Simplify the expression: \(\mathbf{v}_{2} = \frac{10}{5}\mathbf{i} - \frac{4}{5}\mathbf{i} - (\frac{-5 + 8}{5})\mathbf{j} = \frac{6}{5}\mathbf{i} + \frac{3}{5}\mathbf{j}\).

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Key Concepts

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

parallel vectors
When we talk about parallel vectors, we mean that two vectors lie along the same direction. This means one vector can be expressed as a scalar multiple of the other. In the given exercise, we are asked to find a vector \(\mathbf{v}_1\) that is parallel to \(\mathbf{w}\). \(\mathbf{v}_1\) can be represented as \(k\mathbf{w}\), where \(k\) is a scalar. This is important in vector decomposition because finding the parallel component helps in dividing a vector into meaningful parts. Understanding parallel vectors helps to see how certain vector components align with a chosen direction, making analysis easier.
orthogonal vectors
Orthogonal vectors are vectors that are perpendicular to each other. Mathematically, two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal if their dot product is zero, i.e., \(\mathbf{a} \cdot \mathbf{b} = 0\). This concept is used in the given exercise when we need to find \(\mathbf{v}_2\) which is orthogonal to \(\mathbf{w}\). This means \(\mathbf{v}_2\) has no component in the direction of \(\mathbf{w}\), making it easier to analyze decomposition. Orthogonal vectors play a vital role in physics and engineering when dealing with forces, directions, and balanced systems.
dot product
The dot product, also known as the scalar product, measures the extent to which two vectors align. It is calculated as the sum of the products of their corresponding components. For instance, in the exercise, \(\mathbf{v} \cdot \mathbf{w}\) and \(\mathbf{w} \cdot \mathbf{w}\) are calculated to determine the parallel and orthogonal components. The dot product formula is \(\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + ... + a_n b_n\). The result is a scalar and can be positive, negative, or zero. Positive dot products mean vectors point in roughly the same direction, negative means opposite directions, and zero implies orthogonality.
projection of a vector
Projection of a vector \(\mathbf{v}\) onto another vector \(\mathbf{w}\) finds the component of \(\mathbf{v}\) that is in the direction of \(\mathbf{w}\). This is represented by the formula: \({\text{proj}_{\mathbf{w}}{\mathbf{v}}} = \frac{\mathbf{v} \cdot {\mathbf{w}}}{\mathbf{w} \cdot {\mathbf{w}}} \mathbf{w}\). In the given exercise, this formula helps find \(\mathbf{v}_1\), the component of \(\mathbf{v}\) parallel to \(\mathbf{w}\). Projection is essential in vector decomposition as it allows us to break a vector into orthogonal parts with respect to a given direction, simplifying analysis in physics and engineering contexts.

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Most popular questions from this chapter

A force of magnitude 700 pounds is required to hold a boat and its trailer in place on a ramp whose incline is \(10^{\circ}\) to the horizontal. What is the combined weight of the boat and its trailer?

(a) find the dot product v \(\cdot \mathbf{w} ;\) (b) find the angle between \(\mathbf{v}\) and \(\mathbf{w} ;\) (c) state whether the vectors are parallel, orthogonal, or neither. $$ \mathbf{v}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-2 \mathbf{j} $$

Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun's rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun's rays. Let the vector \(\mathbf{A}\) represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure. The total number of watts collected by the panel is given by \(W=|\mathbf{I} \cdot \mathbf{A}|\) Suppose that \(\mathbf{I}=\langle-0.02,-0.01\rangle\) and \(\mathbf{A}=\langle 300,400\rangle\) (a) Find \(\|\mathbf{I}\|\) and \(\|\mathbf{A}\|,\) and interpret the meaning of each. (b) Compute \(W\) and interpret its meaning. (c) If the solar panel is to collect the maximum number of watts, what must be true about I and \(\mathbf{A}\) ?

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