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Use the law of cosines to prove that isometries preserve angles; that is suppose that T : R2 → R2 is an isometry and let P, Q, R E R2 be three noncollinear points in the plane. Denote the images...
#3 please Isometries 1. (Apt) Prove that the composition of two isometries is an isometry. 2.4pt)...
#3 please
Isometries 1. (Apt) Prove that the composition of two isometries is an isometry. 2.4pt) Prove that if F and G is an isometry of the plane and F(AABC) = G(AABC) = AA'B'C', then F = G. That is, an isometry is uniquely determined by three non-collinear points (a triangle) and their images. Consider using an inverse of one of the isometries. 3. (4pt) To prove that a reflection, Rom, is an isometry, it must be shown that it...
step by step please. 30. Let p and q denote quaternions and let a,b E R....
step by step please.
30. Let p and q denote quaternions and let a,b E R. Show that (b) (ap + bq)apbq (c) N(q) = qq* = qq (d) pq)* = q*p* [Hint: First show that (iq)* =-qi, (jq)* =- (kq)* =- (b).] (e) N(pq) -Np)N() [Hint: (c) and (d).] of k. q J, and g K, and then use
2. Let P(2, 2), Q(3,5) and R(4,3) be three points in the xy plane. (a) Determine...
2. Let P(2, 2), Q(3,5) and R(4,3) be three points in the xy plane. (a) Determine the vectors a = PO, b =PŘ, and c =RQ and draw them on the xy plane below. (b) Calculate a - b and compare it to c. What do you notice? Explanation? R 2
Consider a point charge q moving arbitrar ily along a trajectory described by vector function of time r (t). The velocity of the charge is thus V(t)- di,(t)/dt. Suppose Q and Q'represent points o...
Consider a point charge q moving arbitrar ily along a trajectory described by vector function of time r (t). The velocity of the charge is thus V(t)- di,(t)/dt. Suppose Q and Q'represent points on the trajectory where the charge is at time t and was at an earlier time t'. Let R(t) F r,(t) be the vector from the charge to the fixed point P as shown in the figure of particle re volume element de r" a) Prove the...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...
Let S denote the sphere x2 y2 2 = 1. Given two points P(1,0,0), (a) Find the distance between P and Q. Lets call this E...
Let S denote the sphere x2 y2 2 = 1. Given two points P(1,0,0), (a) Find the distance between P and Q. Lets call this Euclidean distance. (b) Find the plane that goes through O, P, Q. What is the intersection of this plane with the sphere? (Hint: use OP × OQ as the the normal vector) (c) Observe that the length of the arc PQ is 0 the angle between OP,0Q in radians. (Hint: You know how to find...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1...
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1 < x2 < X3. Suppose that y = ax + by + c is a parabola passing through the three points (x1, yı), (x2, y), and (x3, Y3). We have that a, b, and c must satisfy i = ax + bx + C V2 = ax + bx2 + c y3 = ax} + bx3 + c Let D = x X2 1....
Read “Instituionalizing our Demise: America vs Multiculturalism” by Roger Kimball on pg 268 and “Reinventing America”...
Read “Instituionalizing our Demise: America vs Multiculturalism” by
Roger Kimball on pg 268 and “Reinventing America” Call for a new
national indentity” by Elizabeth Martinez on pg 275. Create a
double entry notebook for each reading selection It should be
atleast five observation and responses.
wric 268 PART 2 essay pro. exactly how and why their authors disagree. Instead of with parties in conflict as mediators do, you will nt of view designed to appeal to both sides, mediatn posing...
#3 please
Isometries 1. (Apt) Prove that the composition of two isometries is an isometry. 2.4pt) Prove that if F and G is an isometry of the plane and F(AABC) = G(AABC) = AA'B'C', then F = G. That is, an isometry is uniquely determined by three non-collinear points (a triangle) and their images. Consider using an inverse of one of the isometries. 3. (4pt) To prove that a reflection, Rom, is an isometry, it must be shown that it...
step by step please.
30. Let p and q denote quaternions and let a,b E R. Show that (b) (ap + bq)apbq (c) N(q) = qq* = qq (d) pq)* = q*p* [Hint: First show that (iq)* =-qi, (jq)* =- (kq)* =- (b).] (e) N(pq) -Np)N() [Hint: (c) and (d).] of k. q J, and g K, and then use
2. Let P(2, 2), Q(3,5) and R(4,3) be three points in the xy plane. (a) Determine the vectors a = PO, b =PŘ, and c =RQ and draw them on the xy plane below. (b) Calculate a - b and compare it to c. What do you notice? Explanation? R 2
Consider a point charge q moving arbitrar ily along a trajectory described by vector function of time r (t). The velocity of the charge is thus V(t)- di,(t)/dt. Suppose Q and Q'represent points on the trajectory where the charge is at time t and was at an earlier time t'. Let R(t) F r,(t) be the vector from the charge to the fixed point P as shown in the figure of particle re volume element de r" a) Prove the...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...
Let S denote the sphere x2 y2 2 = 1. Given two points P(1,0,0), (a) Find the distance between P and Q. Lets call this Euclidean distance. (b) Find the plane that goes through O, P, Q. What is the intersection of this plane with the sphere? (Hint: use OP × OQ as the the normal vector) (c) Observe that the length of the arc PQ is 0 the angle between OP,0Q in radians. (Hint: You know how to find...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1 < x2 < X3. Suppose that y = ax + by + c is a parabola passing through the three points (x1, yı), (x2, y), and (x3, Y3). We have that a, b, and c must satisfy i = ax + bx + C V2 = ax + bx2 + c y3 = ax} + bx3 + c Let D = x X2 1....
Read “Instituionalizing our Demise: America vs Multiculturalism” by
Roger Kimball on pg 268 and “Reinventing America” Call for a new
national indentity” by Elizabeth Martinez on pg 275. Create a
double entry notebook for each reading selection It should be
atleast five observation and responses.
wric 268 PART 2 essay pro. exactly how and why their authors disagree. Instead of with parties in conflict as mediators do, you will nt of view designed to appeal to both sides, mediatn posing...
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