EBK PRECALCULUS W/LIMITS
EBK PRECALCULUS W/LIMITS
4th Edition
ISBN: 9781337516853
Author: Larson
Publisher: CENGAGE CO
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Chapter 11.3, Problem 43E
(a)

To determine

To verify: The points A(3,21),B(2,2,3),C(3,5,2),D(2,5,4) are the vertices of the parallelogram.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given information:

The points A(3,21),B(2,2,3),C(3,5,2),D(2,5,4) .

Formula used:

Four vertices A, B, C, and D are said to be vertices of parallelogram ABCD if CD=AB and CB=AD that is AB parallel to CD and AD is parallel to CB . Therefore, AB and AD are adjacent sides of parallelogram.

Proof:

Consider the points A(3,21),B(2,2,3),C(3,5,2),D(2,5,4) .

Recall that four vertices A, B, C, and D are said to be vertices of parallelogram ABCD if CD=AB and CB=AD that is AB parallel to CD and AD is parallel to CB . Therefore, AB and AD are adjacent sides of parallelogram.

Now, the value of AB is evaluated as, subtract the coordinate of B from A and write it in vector notation as,

  AB=(23)i+(22)j+(3+1)k=5i+0j2k

Similarly, the vector BC written it in vector notation as,

  BC=(3+2)i+(52)j+(2+3)k=5i+3j+k

Similarly, the vector CD written it in vector notation as,

  CD=(23)i+(55)j+(4+2)k=5i+0j2k

Similarly, the vector DA written it in vector notation as,

  DA=(3+2)i+(25)j+(1+4)k=5i3j+3k

Similarly, the vector AC written it in vector notation as,

  AC=(33)i+(52)j+(2+1)k=0i+3jk

Similarly, the vector BD written it in vector notation as,

  BD=(2+2)i+(52)j+(4+3)k=0i+3jk

It is observed that,

  AB=CDAC=BD

It can be said that, AB is parallel to CD and AC is parallel to BD .

Thus, the quadrilateral formed by the points A(3,21),B(2,2,3),C(3,5,2),D(2,5,4) represent a parallelogram.

Hence, verified.

(b)

To determine

To calculate:The area of the parallelogram.

(b)

Expert Solution
Check Mark

Answer to Problem 43E

The area of the parallelogramis 286 .

Explanation of Solution

Given information:

The points A(3,21),B(2,2,3),C(3,5,2),D(2,5,4) .

Formula used:

Let a×b represents the vector x such that x=x1i+x2j+x3k then the magnitude of the vector x is given by x=x12+x22+x33 .

If there are two three-dimensional vectors in a space given in the form of their components, say, a=a1,a2,a3 and b=b1,b2,b3 , then the cross product of two vectors is defined as ,

  a×b=ijka1a2a3b1b2b3(a2b3a3b2 )i(a1b3a3b1)j+(a1b2a2b1)k

If a=a1,a2,a3 and b=b1,b2,b3 are the adjacent sides of a parallelogram then area of the parallelogram is a×b .

Calculation:

Consider the points A(3,21),B(2,2,3),C(3,5,2),D(2,5,4) .

Recall that four vertices A, B, C, and D are said to be vertices of parallelogram ABCD if CD=AB and CB=AD that is AB parallel to CD and AD is parallel to CB . Therefore, AB and AD are adjacent sides of parallelogram.

Now, the value of AB is evaluated as, subtract the coordinate of B from A and write it in vector notation as,

  AB=(23)i+(22)j+(3+1)k=5i+0j2k

Similarly, the vector BC written it in vector notation as,

  BC=(3+2)i+(52)j+(2+3)k=5i+3j+k

Similarly, the vector CD written it in vector notation as,

  CD=(23)i+(55)j+(4+2)k=5i+0j2k

Similarly, the vector DA written it in vector notation as,

  DA=(3+2)i+(25)j+(1+4)k=5i3j+3k

Similarly, the vector AC written it in vector notation as,

  AC=(33)i+(52)j+(2+1)k=0i+3jk

Similarly, the vector BD written it in vector notation as,

  BD=(2+2)i+(52)j+(4+3)k=0i+3jk

It is observed that,

  AB=CDAC=BD

It can be said that, AB is parallel to CD and AC is parallel to BD .

Now, side AC is adjacent to side AB . So, area of parallelogram is, AC×AB .

Recall that if a=a1,a2,a3 and b=b1,b2,b3 are the adjacent sides of a parallelogram then area of the parallelogram is a×b .

Also, if there are two three-dimensional vectors in a space given in the form of their components, say, a=a1,a2,a3 and b=b1,b2,b3 , then the cross product of two vectors is defined as ,

  a×b=ijka1a2a3b1b2b3(a2b3a3b2 )i(a1b3a3b1)j+(a1b2a2b1)k

Compute the value, AC×AB ,

  AC×AB=ijk031502=(6)i(5)j+(15)k=6i+5j+15k

And, magnitude of AC×AB is,

  AC×AB=(6)2+(5)2+(15)2=286

Thus, the area of the parallelogramis 286 .

Chapter 11 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 11.1 - Prob. 11ECh. 11.1 - Prob. 12ECh. 11.1 - Prob. 13ECh. 11.1 - Prob. 14ECh. 11.1 - Prob. 15ECh. 11.1 - Prob. 16ECh. 11.1 - Prob. 17ECh. 11.1 - Prob. 18ECh. 11.1 - Prob. 19ECh. 11.1 - Prob. 20ECh. 11.1 - Prob. 21ECh. 11.1 - Prob. 22ECh. 11.1 - Prob. 23ECh. 11.1 - Prob. 24ECh. 11.1 - Prob. 25ECh. 11.1 - Prob. 26ECh. 11.1 - Prob. 27ECh. 11.1 - Prob. 28ECh. 11.1 - Prob. 29ECh. 11.1 - Prob. 30ECh. 11.1 - Prob. 31ECh. 11.1 - Prob. 32ECh. 11.1 - Prob. 33ECh. 11.1 - Prob. 34ECh. 11.1 - Prob. 35ECh. 11.1 - Prob. 36ECh. 11.1 - Prob. 37ECh. 11.1 - Prob. 38ECh. 11.1 - Prob. 39ECh. 11.1 - Prob. 40ECh. 11.1 - Prob. 41ECh. 11.1 - Prob. 42ECh. 11.1 - Prob. 43ECh. 11.1 - Prob. 44ECh. 11.1 - Prob. 45ECh. 11.1 - Prob. 46ECh. 11.1 - Prob. 47ECh. 11.1 - Prob. 48ECh. 11.1 - Prob. 49ECh. 11.1 - Prob. 50ECh. 11.1 - Prob. 51ECh. 11.1 - Prob. 52ECh. 11.1 - Prob. 53ECh. 11.1 - Prob. 54ECh. 11.1 - Prob. 55ECh. 11.1 - Prob. 56ECh. 11.1 - Prob. 57ECh. 11.1 - Prob. 58ECh. 11.1 - Prob. 59ECh. 11.1 - Prob. 60ECh. 11.1 - 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