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Chapter 10: Q.34 (page 849)

Find the volume of the parallelepiped determined by \(u=<2,4,-1>\), \(v=<0,-3,2>\), and \(w=<-1,1,5>\).

Short Answer

Expert verified

The volume of the parallelepied is \(39\) cu units.

Step by step solution

01

Given Information

The given vectors are \(u=<2,4,-1>\), \(v=<0,-3,2>\), and \(w=<-1,1,5>\).

The volume of the parallelepiped is \(|(u\times v)\cdot w|\).

02

Find the cross product of two vector

First, find the \(u\timesv\).

\(\begin{align*}u\times v&= \begin{vmatrix}i&j&k\\2&4&-1\\0&-3&2\end{vmatrix}\\&=5i-4j-6k\end{align*}\)

03

Find the volume

\(\begin{align*}V&=|(5i-4j-6k)\cdot (-i+j+5k)|\\&=|-5-4-30|\\&=39\end{align*}\)

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Most popular questions from this chapter

In Exercises 24-27, find compuv, projuv, and the component of v orthogonal tou.

u=1,2,v=3,5

Find u+vandu-valso sketch u,v,u+v,u-v

role="math" localid="1649603034674" u=1,-4,6andv=2,-4,7

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.)

If u, v and w are three vectors in 鈩�3, which of the following products make sense and which do not?

localid="1649346164463" (a)u路(v路w)(b)u路(v脳w)(c)u脳(v路w)(d)u脳(v脳w)

What is meant by the triangle determined by vectors u and v in 鈩�3? How do you find the area of this triangle?
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