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Chapter 1: Q1.3P (page 7)

Find the angle between the body diagonals of a cube.

Short Answer

Expert verified

The angle between the digonals is 70.52鈭�.

Step by step solution

01

Explain the concept and draw the cube using given information

To find the angle between the diagonals, the vector diagonals must be evaluated and then the dot formula must be used.

The cube has 1 unit side and one of the corner is coinciding with the origin. using this information, the cube is drawn as follows:

02

Assume the vectors

The principle diagonals are OM鈫�and BA鈫�. The corrdinates of points O, M, A and B are O=0,0,0,M=1,1,1,B=0,1,0 and A=1,0,1.

Find the vectors OM鈫�and BA鈫�.

OM鈫�=1-0i+1-0j+1-0k=i+j+k

Solve the vector BA鈫�.

BA鈫�=(1-0)i+(0-1)j+(1-0)k=i-j+k

03

Find the dot product between OM→ and AB→ .

The formula of the dot product of the vectors OM鈫�and BA鈫�is

OM鈫�BA鈫�=OMBAcos胃,胃is the angle between the vectos OM鈫�and BA鈫�.

Find the dot product of the vectors OM鈫�and BA鈫�.

OM鈫�.BA鈫�=OMBAcos胃2=3cos胃cos胃=23胃=70.52鈭�

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Most popular questions from this chapter

Suppose that f is a function of two variables (y and z) only. Show that the gradient 鈭�f=(鈭�f/鈭�y)y^(鈭�f/鈭�z)z^transforms as a vector under rotations, Eq 1.29. [Hint: (鈭�f/鈭�y炉)=(鈭�f/鈭�y)(鈭�f/鈭�y炉)+(鈭�f/鈭�z)(鈭�z/鈭�y炉),and the analogous formula for 鈭�f/鈭�z炉. We know that localid="1654595255202" y炉=测肠辞蝉蠒+锄蝉颈苍蠒and z=-ycos鈭�+zcos鈭�;鈥漵olve鈥� these equations for y and z (as functions of localid="1654325243865" y炉and z(as functions of yand z), and compute the needed derivatives 鈭�f/鈭�y,鈭�z/鈭�y, etc]

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