Consider the lines P and Q in ℝ 2 in the accompanying figure. Consider the linear transformation T ( x → ) = ref Q ( ref p ( x → ) ) ; that is, we first reflect x → about P andthen we reflect the result about Q . a. For the vector x → given in the figure, sketch T ( x → ) .What angle do the vectors x → and T ( x → ) enclose?What is the relationship between the lengths of x → and T ( x → ) ? b. Use your answer in part (a) to describe the transformation T geometrically, as a reflection, rotation,shear, or projection. c. Find the matrix of T . d. Give a geometrical interpretation of the linear transformation L ( x → ) = ref P ( ref Q ( x → ) ) , and find thematrix of L.
Consider the lines P and Q in ℝ 2 in the accompanying figure. Consider the linear transformation T ( x → ) = ref Q ( ref p ( x → ) ) ; that is, we first reflect x → about P andthen we reflect the result about Q . a. For the vector x → given in the figure, sketch T ( x → ) .What angle do the vectors x → and T ( x → ) enclose?What is the relationship between the lengths of x → and T ( x → ) ? b. Use your answer in part (a) to describe the transformation T geometrically, as a reflection, rotation,shear, or projection. c. Find the matrix of T . d. Give a geometrical interpretation of the linear transformation L ( x → ) = ref P ( ref Q ( x → ) ) , and find thematrix of L.
Solution Summary: The author explains the relationship between the lengths of stackreltox and T(
Consider the lines P and Q in
ℝ
2
in the accompanying figure. Consider the linear transformation
T
(
x
→
)
=
ref
Q
(
ref
p
(
x
→
)
)
; that is, we first reflect
x
→
about P andthen we reflect the result about Q.
a. For the vector
x
→
given in the figure, sketch
T
(
x
→
)
.What angle do the vectors
x
→
and
T
(
x
→
)
enclose?What is the relationship between the lengths of
x
→
and
T
(
x
→
)
? b. Use your answer in part (a) to describe the transformation T geometrically, as a reflection, rotation,shear, or projection. c. Find the matrix of T. d. Give a geometrical interpretation of the linear transformation
L
(
x
→
)
=
ref
P
(
ref
Q
(
x
→
)
)
, and find thematrix of L.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
A point A(-5, -3) is translated by the
vector v = and then reflected about
the line x = 0. Find A" after the second
transformation.
. Let A= (2, 3, 1) and B= (4, -1, 2) and C = (6, 2, 3). Find the equation of the line in parametric form
passing through C that intersects AB is perpendicular to AB .
4.
Consider the points A(1, –1, 4), B(2, – 2, 5) and O(0, 0, 0).
(a)
Calculate the cosine of the angle between OA and AB.
(b) Find a vector equation of the line L1 which passes through A and B.
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY