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Chapter 10: Problem 20

Find \(\operatorname{com} p_{b}\) a and projo a. \(\mathbf{a}=3 \mathbf{i}+\mathbf{j}, \mathbf{b}=4 \mathbf{i}-3 \mathbf{j}\)

Short Answer

Expert verified
The projection of vector a onto vector b is 0.36 and the component of vector a along b is \(1.44\mathbf{i} - 1.08\mathbf{j}\)

Step by step solution

01

Calculate the Dot Product of a and b

The dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is calculated using their components. In this case, \(\mathbf{a} . \mathbf{b}=a_{i} \cdot b_{i}+a_{j} \cdot b_{j}=3\cdot4+1\cdot(-3)=12-3=9\)
02

Calculate the Magnitude of b

The magnitude of vector \(\mathbf{b}\) is given by \(\sqrt{b_{i}^{2}+b_{j}^{2}}=\sqrt{4^2+(-3)^2}=\sqrt{16+9}=\sqrt{25}=5\)
03

Compute the projection

The projection of \(\mathbf{a}\) onto \(\mathbf{b}\) uses the dot product of \(\mathbf{a}\) and \(\mathbf{b}\) and the square of the magnitude of \(\mathbf{b}\). Thus \(\operatorname{projo}_{\mathbf{b}} \mathbf{a}= (\mathbf{a} . \mathbf{b}) / |\mathbf{b}|^2=(9)/(5^2)=(9)/(25)=0.36\)
04

Compute the component of a

\(\operatorname{com}_{\mathbf{b}} \mathbf{a} = \operatorname{projo}_{\mathbf{b}} \mathbf{a} \cdot \mathbf{b} =0.36 \cdot( 4\mathbf{i}-3 \mathbf{j})=1.44 \mathbf{i}- 1.08 \mathbf{j}\)

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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 when working with vectors. It combines two vectors and results in a scalar. To find the dot product of two vectors, such as \(\mathbf{a} = 3\mathbf{i} + \mathbf{j}\) and \(\mathbf{b} = 4\mathbf{i} - 3\mathbf{j}\), you multiply their corresponding components and sum the results.
For these vectors, the dot product \(\mathbf{a} \cdot \mathbf{b}\) is calculated as follows:
  • Multiply the \(\mathbf{i}\)-components: \(3 \cdot 4 = 12\).
  • Multiply the \(\mathbf{j}\)-components: \(1 \cdot (-3) = -3\).
  • Add these two results together: \(12 - 3 = 9\).
So, the dot product of \(\mathbf{a}\) and \(\mathbf{b}\) is \(9\). The dot product is especially useful because it helps in determining the angle between vectors and projections.
Magnitude of a Vector
Magnitude, or the length of a vector, is a measurement of how long the vector is. It's calculated using the Pythagorean theorem in the context of vector components. For a vector \(\mathbf{b} = 4\mathbf{i} - 3\mathbf{j}\), the magnitude is derived from its components.
The formula to find the magnitude \(|\mathbf{b}|\) is:
  • Square each component: \(4^2 = 16\) and \((-3)^2 = 9\).
  • Sum these squared values: \(16 + 9 = 25\).
  • Take the square root of this sum: \(\sqrt{25} = 5\).
So the magnitude of \(\mathbf{b}\) is \(5\). This concept is crucial for normalizing vectors and calculating projections.
Vector Components
Vector components allow us to break down a vector into its base parts. This breakdown is essential for calculations, such as the dot product and finding projections. A vector like \(\mathbf{a}=3 \mathbf{i}+ \mathbf{j}\) has components:
  • In the \(\mathbf{i}\) direction: it has a component of \(3\).
  • In the \(\mathbf{j}\) direction: it has a component of \(1\).
By identifying these components, you can easily perform various operations. They also allow representation of vectors in coordinate systems, aiding in both physical and mathematical applications.
Projection of Vectors
The projection of one vector onto another serves to "project" a vector in the direction of another. This provides a vector parallel to the second vector. In the case of vectors \(\mathbf{a}\) onto \(\mathbf{b}\), the projection is computed using the dot product and the magnitude of the vector \(\mathbf{b}\).
The formula used here is:
  • Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{b}\): \(9\).
  • Determine the square of the magnitude of \(\mathbf{b}\): \(5^2 = 25\).
  • Divide the dot product by this square: \(\frac{9}{25} = 0.36\).
This gives a scalar that is then multiplied by \(\mathbf{b}\) to find the actual vector projection:
  • \(0.36 \cdot (4\mathbf{i}-3\mathbf{j}) = 1.44 \mathbf{i} - 1.08 \mathbf{j}\).
Projections are vital in physics and engineering, especially in calculating forces and motions along particular directions.

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Most popular questions from this chapter

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