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Chapter 10: Q.24 (page 848)

Compute the lengths of the four diagonals of the parallelepiped

determined by \(u=i\), \(v=2j\), and \(w=2k\).

Short Answer

Expert verified

The lengths of the four diagonals are \(5,5,2\sqrt2,\text{ and }3\).

Step by step solution

01

Given Information

A parallelepiped formed by \(u=i\), \(v=2j\), and \(w=2k\).

There are four diagonals three faces and one body diagonal.

Three faces diagonals are:

\(d_1=u+v\)

\(d_2=u+w\)

\(d_3=v+w\)

One body diagonal is:

\(d_4=u+v+w\)

02

Calculate the lengths of the diagonals

The length of 1st face diagonal:

\(d_1=|i+2j|\)

\(d_1=\sqrt{1+4}\)

\(d_1=\sqrt5\)

The length of 2nd face diagonal:

\(d_2=|i+2k|\)

\(d_2=\sqrt{1+4}\)

\(d_2=\sqrt5\)

The length of 3rd face diagonal:

\(d_3=|2j+2k|\)

\(d_3=\sqrt{4+4}\)

\(d_3=2\sqrt2\)

The length of the body diagonal:

\(d_4=|i+2j+2k|\)

\(d_4=\sqrt{1+4+4}\)

\(d_4=3\)

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Most popular questions from this chapter

Prove the first part of Theorem 1.31(a): If k>0, then limx鈫�鈭�xk=鈭�. (Hint: Given M>0, choose N=M1/k. Then show that for x>N it must follow that xk>M.)

In Exercises 30鈥�35 compute the indicated quantities when u=(鈭�3,1,鈭�4),v=(2,0,5),andw=(1,3,13)

Find the area of the parallelogram determined by the vectors u and v.

If u and v are nonzero vectors in 鈩�3, what is the geometric relationship between u,v and u脳v?

Find PQ鈫�

P=-1,5,4,Q=-1,6,4

In Exercises 36鈥�41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(4,鈭�2),Q(鈭�2,0),R(1,鈭�5)

(Hint: Think of the xy-plane as part of 鈩�3.)
See all solutions

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