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Chapter 10: Problem 3

Compute, \(a+b, a-2 b, 3 a\) and \(\|5 b-2 a\|.\) $$\mathbf{a}=\langle 2,4\rangle, \mathbf{b}=\langle 3,-1\rangle$$

Short Answer

Expert verified
\(a + b = \langle 5, 3 \rangle\), \(a - 2b = \langle -4, 6 \rangle\), \(3a = \langle 6, 12 \rangle\), \( \|\|5b-2a\|\| = \sqrt{290}\).

Step by step solution

01

Compute \(a + b\)

Add the corresponding components of vectors \(a\) and \(b\) together. This gives us \(\langle 2+3, 4+(-1) \rangle = \langle 5, 3 \rangle \).
02

Compute \(a - 2b\)

First, multiply vector \(b\) by 2 to get \(2b = \langle 2*3, 2*(-1) \rangle = \langle 6, -2 \rangle\). Then subtract the resulting vector from \(a\) to get \(a - 2b = \langle 2-6, 4-(-2) \rangle = \langle -4, 6 \rangle\).
03

Compute \(3a\)

Multiply each component of the vector \(a\) by 3 to get \(3a = \langle 3*2, 3*4 \rangle = \langle 6, 12 \rangle.\)
04

Compute \(\|5b - 2a\|\)

First, compute \(5b = \langle 5*3, 5*(-1) \rangle = \langle 15, -5 \rangle\) and \(2a = \langle 2*2, 2*4 \rangle = \langle 4, 8 \rangle \). Then subtract to get \(5b - 2a = \langle 15-4, -5-8 \rangle = \langle 11, -13 \rangle\). Finally, to calculate the norm of this vector, use the formula \(\sqrt{(11^2 + (-13)^2)} = \sqrt{121 + 169} = \sqrt{290}\).

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Key Concepts

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

Vectors
Vectors are mathematical objects used to represent quantities that have both magnitude and direction. Think of a vector as an arrow pointing from one point to another in space. Each vector has components that describe its direction and length. For example, vector \(\mathbf{a} = \langle 2, 4 \rangle\) has components 2 in the x-direction and 4 in the y-direction. These components are usually written in angle brackets or sometimes as a column in matrices.
  • Components: The individual values that define the vector along each axis. In 2D, this is usually x and y; in 3D, it would be x, y, and z.
  • Notation: Vectors are often represented by bold type (\(\mathbf{a}\), \(\mathbf{b}\)) or an arrow on top if handwritten (\(\vec{a}\)).
Understanding vectors is crucial for fields like physics and engineering because they describe quantities like force, velocity, and displacement which need both size and direction.
Vector Addition
Vector addition involves combining two or more vectors to form a new vector, known as the resultant vector. This is similar to adding individual numbers but done for each component separately. For example, to add vectors \(\mathbf{a} = \langle 2, 4 \rangle\) and \(\mathbf{b} = \langle 3, -1 \rangle\), you add the x-components together and the y-components together: \(\mathbf{a} + \mathbf{b} = \langle 2+3, 4+(-1) \rangle = \langle 5, 3 \rangle\).
  • Procedure: Done component-wise, taking each corresponding component of the vectors and adding them.
  • Resultant Vector: The new vector formed from the addition, expressing the combined effect of the original vectors.
Vector addition is useful in various applications such as determining the total displacement of an object or the resultant force acting on a body.
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the magnitude of the vector without changing its direction. It affects each component of the vector uniformly. For instance, multiplying vector \(\mathbf{a} = \langle 2, 4 \rangle\) by 3 gives \(3\mathbf{a} = \langle 3\times2, 3\times4 \rangle = \langle 6, 12 \rangle\).
  • Effect: The vector's magnitude is scaled by the absolute value of the scalar, while its direction remains unchanged unless the scalar is negative, which reverses the direction.
  • Examples: Useful in physics where vectors like velocity are scaled by time or acceleration factors.
This operation is crucial when adjusting vectors for various mathematical or physical computations, such as changing scales or units.
Vector Subtraction
Vector subtraction is the process of finding the difference between two vectors. It's similar to vector addition, except you subtract each corresponding component of the vectors. For example, finding \(\mathbf{a} - 2\mathbf{b}\) where \(\mathbf{b}\) is multiplied by 2 first, involves calculating \(2\mathbf{b} = \langle 6, -2 \rangle\) and then \(\mathbf{a} - 2\mathbf{b} = \langle 2 - 6, 4 - (-2) \rangle = \langle -4, 6 \rangle\).
  • Procedure: Subtract each component of the second vector from the corresponding component of the first vector.
  • Result: A new vector that represents the directional and magnitude change from the second to the first vector.
Subtracting vectors helps in determining relative motion or displacement in physics and is fundamental in calculations involving difference vectors.

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Most popular questions from this chapter

Find an equation of the plane containing the lines \(\left\\{\begin{array}{l}x=1-t \\ y=2+3 t \\ z=2 t\end{array} \quad \text { and } \quad\left\\{\begin{array}{l}x=1-s \\ y=5 \\ z=4-2 s\end{array}\right.\right.\)

Suppose that in a particular county, ice cream sales (in thousands of gallons) for a year is given by the vector \(\mathbf{s}=\langle 3,5,12,40,60,100,120,160,110,50,10,2\rangle .\) That is, 3000 gallons were sold in January, 5000 gallons were sold in February, and so on. In the same county, suppose that murders for the year are given by the vector \(\mathbf{m}=\langle 2,0,1,6,4,8,10,13,8,2,0,6\rangle .\) Show that the aver- age monthly ice cream sales is \(\bar{s}=56,000\) gallons and that the average monthly number of murders is \(\bar{m}=5 .\) Compute the vectors a and \(\mathbf{b},\) where the components of a equal the components of s with the mean 56 subtracted (so that \(\mathbf{a}=\langle-53,-51,-44, \ldots\rangle)\) and the components of \(\mathbf{b}\) equal the components of \(\mathrm{m}\) with the mean 5 subtracted. The correlation between ice cream sales and murders is defined as \(\rho=\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|} .\) Often, a positive correlation is incorrectly interpreted as meaning that a "causes" b. (In fact, correlation should never be used to infer a cause-and-effect relationship.) Explain why such a conclusion would be invalid in this case.

Use the parallelepiped volume formula to determine whether the vectors are coplanar. $$\langle 1,0,-2\rangle,\langle 3,0,1\rangle \text { and }\langle 2,1,0\rangle$$

Determine whether the lines are parallel, skew or intersect. $$\left\\{\begin{array}{ll} x=1-2 t & \\ y=2 t & \text { and } \\ z=5-t & \end{array}\left\\{\begin{array}{l} x=3+2 s \\ y=-2 \\ z=3+2 s \end{array}\right.\right.$$

For the Mandelbrot set and associated Julia sets, functions of the form \(f(x)=x^{2}-c\) are analyzed for various constants \(c\) The iterates of the function increase if \(\left|x^{2}-c\right|>|x|\). Show that this is true if \(|x|>\frac{1}{2}+\sqrt{\frac{1}{4}+c}\)
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