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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∘.

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→and BA→. 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→and BA→.

OM→=1-0i+1-0j+1-0k=i+j+k

Solve the vector BA→.

BA→=(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→and BA→is

OM→BA→=OMBAcosθ,θis the angle between the vectos OM→and BA→.

Find the dot product of the vectors OM→and BA→.

OM→.BA→=OMBAcosθ2=3cosθcosθ=23θ=70.52∘

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Most popular questions from this chapter

Although the gradient, divergence, and curl theorems are the fundamental integral theorems of vector calculus, it is possible to derive a number of corollaries from them. Show that:

(a)∫v∇Tdτ=∮sT da. [Hint:Let v = cT, where c is a constant, in the divergence theorem; use the product rules.]

(b)∫v∇×vdτ=∮sv× da. [Hint:Replace v by (v x c) in the divergence

theorem.]

(c)∫vT∇2U+∇T⋅∇Udτ=∮sT∇U ⋅da . [Hint:Let in the

divergence theorem.]

(d)∫vU∇2T+∇U⋅∇Vdτ=∮sU∇T ⋅da. [Comment:This is sometimes

called Green's second identity; it follows from (c), which is known as

Green's identity.]

(e) ∫S∇T×da=∮PT ⋅dl[Hint:Let v = cT in Stokes' theorem.]

Test the divergence theorem for the function v=(xy)i+(2yz)j+(3zx)k..Take as your volume the cube shown in Fig. 1.30, with sides of length 2.

Calculate the curls of the vector functions in Prob. 1.15.

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(c) The direct straight line.

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Check the divergence theorem for the function

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