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Chapter 3: Q1P (page 99)

Draw diagrams and prove (4.1).

Short Answer

Expert verified

The diagram is verified using (4.1) theorem.

Step by step solution

01

Formula for the magnitude of the components of the vector:

If Ax,Ay,and Azare the magnitudes of components of vectorA→along the coordinate axis, then A→=Axi^+Ayj^+Azk^, wherei^,j^,k^respectively are the unit vectors along x,yand zaxis.

02

The diagram of vector A→  in two dimensions:

Using Pythagoras theorem, the length of A , named A=|A|is given by,

A2=Ax2+Ay2A=Ax2+Ay2

Where the dotted lines represent the components of the vector Ax and Ay

03

The diagram of vector A→  in three dimensions:

Using Pythagoras theorem in 3D, the length of A , named A=|A|is given by,

A2=Ax2+Ay2+Az2A=Ax2+Ay2+Az2

Where the dotted lines represent the components of the vectorAx,Ayand Az.

Hence, the condition is proved.

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Most popular questions from this chapter

Let each of the following matricesM describe a deformation of the ( x , y)plane for each given Mfind: the Eigen values and eigenvectors of the transformation, the matrix Cwhich Diagonalizes Mand specifies the rotation to new axes(x',y')along the eigenvectors, and the matrix D which gives the deformation relative to the new axes. Describe the deformation relative to the new axes.

(2-1-12)

Find the distance between the two given lines.

x-12=y+23=2z-14andx+2-1=2-y2,z=12

For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using.

6.x+y-z=13x+2y-2z=3

Find a vector perpendicularto both i+j and i-2k .

Evaluate the determinants in Problems 1 to 6 by the methods shown in Example 4. Remember that the reason for doing this is not just to get the answer (your computer can give you that) but to learn how to manipulate determinants correctly. Check your answers by computer.

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