Planes_ Attempt review _ eClass

3/29/24, 1:37 PM Planes: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=14610286&cmid=7601844 1/8 Started on Friday, 19 January 2024, 2:56 PM State Finished Completed on Sunday, 21 January 2024, 12:36 PM Time taken 1 day 21 hours Grade 42.00 out of 42.00 ( 100 %)

3/29/24, 1:37 PM Planes: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=14610286&cmid=7601844 2/8 Question 1 Correct Mark 7.00 out of 7.00 Planes In 3-dimensional space , a 2-dimensional plane is a flat surface. This can be made precise mathematically using vectors. Moreover, using linear algebra, it is possible to describe what is a 2-dimensional plane (or just a plane) in . For an introduction to planes, in particular to the direction vectors and that appear below, you can watch the video Planes, part 1 . (In the picture above, the notation is used for the vector .) If , a plane in takes the form where are vectors in , and where are not scalar multiples of each other. The equation describing the position vector of a general point on is called a vector equation for . The vectors and are parallel to and are called direction vectors. is the position vector of a point on the plane . It is explained in the video Planes, part 2 what is the meaning of the vector equation describing a plane. The equation defines a plane; true or false?  . The equation defines a plane; true or false?  . The equation defines a plane; true or false?  . Example Consider the plane with vector equation . This equation defines a plane because  . The following point is in :  . This is because we may take in the above vector equation for . Another point on is , which we obtain by taking 1  and -2  . true false false (A) the vectors [1,1,1] and [0,1,2] are not scalar multiples of each other (2,-1,1) Correct

3/29/24, 1:37 PM Planes: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=14610286&cmid=7601844 4/8 Question 3 Correct Mark 9.00 out of 9.00 Finding a vector equation for a plane through 3 points Given 3 points in , there is a unique plane containing all of them. When , this means that there is a unique book that you can hold with 3 different fingers. If are the position vectors of 3 points on a plane, and if these points are not all on the same line, then a vector equation for is Taking 0  and 0  verifies that is on . Taking 1  and 0  verifies that is on . Taking 0  and 1  verifies that is on . Example Using the above formula, the plane through the three points with position vectors is [1,2,3,4]  [0,-2,-2,-4]  [1,0,-2,-5]  . (To enter vectors, use square brackets, commas and no spaces.) Correct Marks for this submission: 9.00/9.00.

3/29/24, 1:37 PM Planes: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=14610286&cmid=7601844 5/8 Question 4 Correct Mark 3.00 out of 3.00 Parametric equations for a plane in 3-dimensional space If a plane in R 3 has vector equation x = p + s v + t w , then a general point ( x 1 , x 2 , x 3 ) on with position vector x = [ x 1 , x 2 , x 3 ] has coordinates satisfying x 1 = p 1 + sv 1 + tw 1 , x 2 = p 2 + sv 2 + tw 2 , x 3 = p 3 + sv 3 + tw 3 , where p = [ p 1 , p 2 , p 3 ], v = [ v 1 , v 2 , v 3 ], w = [ w 1 , w 2 , w 3 ]. These equations for x 1 , x 2 , x 3 are called the parametric equations of . For example, if has vector equation x = [1, 0, − 1] + s [1, 2, 3] + t [ − 1, 1, 2] , then it has parametric equations x 1 = 1 + s − t , x 2 = 2 s + t , x 3 = − 1 + 3 s + 2 t . We can reverse this process to obtain a vector equation from parametric equations. For example, if a plane has parametric equations x 1 = 5 + 2 s + 3 t , x 2 = − 4 + 6 s + 8 t , x 3 = 10 + 11 s − 12 t , then a vector equation for the line is x = [5,-4,10]  + s [2,6,11]  + t [3,8,-12]  . (To enter vectors, use square brackets, commas and no spaces.) Correct Marks for this submission: 3.00/3.00.

3/29/24, 1:37 PM Planes: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=14610286&cmid=7601844 7/8 Question 6 Correct Mark 5.00 out of 5.00 How to find the general equation of a plane in \(\mathbb{R}^3\) with vector equation \ (\mathbf{x}=\mathbf{p}+s\mathbf{v}+t\mathbf{w}\): 1. Find a non-zero vector \(\mathbf{n}=[a,b,c]\) that is orthogonal to both \(\mathbf{v}\) and \(\mathbf{w}\). 2. Expand the equation in normal form with position vector \(\mathbf{p}\) and normal vector \(\mathbf{n}\). An example about how to find a non-zero vector \(\mathbf{n}=[a,b,c]\) that is orthogonal to two given direction vectors is presented in the second half (starting at 7:22) of the video Planes, part 3 . Example The plane \(\mathbf{x}=[1,3,2]+s[4,-2,0]+t[1,-1,1]\) has normal vector \(\mathbf{n}=[2,\) 4  \(,\) 2  \(]\), so a general equation for the plane is \(\qquad 2x+\) 4  \(y+\) 2  \(z=\) 18  . The general equation of a plane is unique up to a non-zero scalar multiple. Correct Marks for this submission: 5.00/5.00.

3/29/24, 1:37 PM Planes: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=14610286&cmid=7601844 8/8 Question 7 Correct Mark 6.00 out of 6.00 How to find a vector equation of a plane with general equation \(ax+by+cz=d\): 1. Choose two non-parallel vectors \(\mathbf{v}\) and \(\mathbf{w}\) that are orthogonal to \(\mathbf{n}=[a,b,c]\). There are many possible choices. 2. Choose a point on the plane with position vector \(\mathbf{p}=[p_1,p_2,p_3]\), so it must satisfy \(ap_1+bp_2+cp_3=d\). There are many to choose from. 3. Then the plane has vector equation \(\mathbf{x}=\mathbf{p}+s\mathbf{v}+t\mathbf{w}\). An example about how to find non-parallel direction vectors \(\mathbf{v}\) and \(\mathbf{w}\) once a normal vector \(\mathbf{n}\) is given is provided in the first half of the video Planes, part 3 . Example The vectors \(\mathbf{v}=[2,1,\) -4  \(]\) and \(\mathbf{w}=[1,1,\) -1  \(]\) are not parallel and they are both orthogonal to \(\mathbf{n}=[3,-2,1]\) and the point \( (1,-2,\) 2  \( )\) is on the plane \(3x-2y+z=9\), so \(\qquad \mathbf{x}=[1,-2,\) 2  \(]+s[2,1,\) -4  \(]+t[1,1,\) -1  \(]\) is a vector equation for the plane \(3x-2y+z=9\). Correct Marks for this submission: 6.00/6.00.