Download the App!
Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.
Question
Answered step-by-step
(a) Find a nonzero vector orthogonal to the plane through the points $ P, Q, $ and $ R $, and (b) find the area of triangle $ PQR $.
$ P (0, 0, -3) $ , $ Q (4, 2, 0) $ , $ R (3, 3, 1) $
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by Dylan Bates
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
03:14
Wen Zheng
Calculus 3
Chapter 12
Vectors and the Geometry of Space
Section 4
The Cross Product
Vectors
Johns Hopkins University
Campbell University
Oregon State University
Boston College
Lectures
02:56
In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.
11:08
In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.
04:25
(a) Find a nonzero vector …
03:44
02:40
08:23
01:21
01:32
01:10
02:27
03:20
01:49
Welcome back to another cross products problem Were given three points in three d. Space and we're asked to find victor Orthogonal to the plane created by those three points. So one of the things we can do is given these three points. We can find two vectors, say PQ and Pr And since we have two vectors in the plane, the cross product between them is going to be perpendicular to that plane. So PQ is just q minus P. That's going to be 4 0. It's for to minus zero, which is two and zero minus negative three, Which is just three. Similarly Pr is ar minus P three minus zero, three minus zero, one minus negative three. We can plug these vectors into our matrix here for two, three and three three, four. And then use the technique outlined in our textbook defined PQ Cross pr So remember we ignore the first column of this matrix And then look at two times 4 And it's three times 3, two times 4 -3 times three I minus. Then we ignore the second column four times four And it's three times 3 it's for -3 times three J. Plus. And then we ignore the third column And we'll look at four times 3 -2 times three three -2 times three. Okay, simplifying this a little bit two times four is eight, three times 3 is nine. So we're looking at negative one in our first coordinate four times 4 is 16, three times 3 is nine, 15 -97. So we're looking at negative seven And then four times 3 is 12, 2 times three is 6, -6. His six of course the negative of this vector would also be perpendicular to the plane. And so either of these answers or any scalar multiples of them will work. Now, what if we wanted to find the area of this triangle peak? You are? Well, it turns out you can do that just by looking at the absolute value of the cross product divided by two. Let's write that out. So the area of P. Q. The area of triangle. There we go. He Q. R. Is just going to be the magnitude of PQ cross pr that leave us the area of a parallelogram. And then we divide that by two to get the area of that triangle. And so this is just going to be the square root of the first component squared plus the second component squared plus the third component squared, All divided by two. And so we're looking at the square root of one plus 49 plus 36 All divided by two, giving us the square root of 86 over to as the area of triangle P Q R. Thanks for watching.
View More Answers From This Book
Find Another Textbook
Baylor University
University of Michigan - Ann Arbor
University of Nottingham
01:22
'At the start of 2010, an online social group had 360 million members S…
02:32
'58. A box is completely filled by 18 unit cubes: What is the volume of…
02:23
'A stereo speaker in the shape of a triangular pyramid has o height of …
01:30
'The number of carbon atoms in fossil is given by the function y 5100(0…
01:16
'positive angle and one negative angle that are coterminal with the giv…
02:21
'The demand for a particular breakfast pastry is given by the demand eq…
02:50
'The graph of an exponential function passes through (2,45) and (4,405)…
02:53
"The radius of a spherical ball is increasing at a rate of 2 cm/min. At…
02:01
'From the top of a 187-ft lighthouse, the angle of depression [I to a s…
06:11
'The radius of a circular disk is given as 18 cm with a maximum error i…