Consider the regular tetrahedron in the accompanying sketch whose center is at the origin. Let v → 0 , v → 1 , v → 2 , v → 3 be the position vectors of the four vertices of the tetrahedron: v → 0 = O P → 0 , ... , v → 3 = O P → 3 . a. Find the sum v → 0 + v → 1 + v → 2 + v → 3 . b. Find the coordinate vector of v → 0 with respect to the basis v → 1 , v → 2 , v → 3 . c. Let T he the linear transformation with T ( v → 0 ) = v → 3 , T ( v → 3 ) = v → 1 , and T ( v → 1 ) = v → 0 . What is T ( v → 2 ) ? Describe the transformation T geometrically (as a reflection, rotation. projection, or whatever). Find the matrix B of T with respect to the basis v → 1 , v → 2 , v → 3 . What is B 3 ? Explain.
Consider the regular tetrahedron in the accompanying sketch whose center is at the origin. Let v → 0 , v → 1 , v → 2 , v → 3 be the position vectors of the four vertices of the tetrahedron: v → 0 = O P → 0 , ... , v → 3 = O P → 3 . a. Find the sum v → 0 + v → 1 + v → 2 + v → 3 . b. Find the coordinate vector of v → 0 with respect to the basis v → 1 , v → 2 , v → 3 . c. Let T he the linear transformation with T ( v → 0 ) = v → 3 , T ( v → 3 ) = v → 1 , and T ( v → 1 ) = v → 0 . What is T ( v → 2 ) ? Describe the transformation T geometrically (as a reflection, rotation. projection, or whatever). Find the matrix B of T with respect to the basis v → 1 , v → 2 , v → 3 . What is B 3 ? Explain.
Solution Summary: The author calculates the sum of stackrelto 'v_0', and the number of vectors.
Consider the regular tetrahedron in the accompanying sketch whose center is at the origin. Let
v
→
0
,
v
→
1
,
v
→
2
,
v
→
3
be the position vectors of the four vertices of the tetrahedron:
v
→
0
=
O
P
→
0
,
...
,
v
→
3
=
O
P
→
3
. a. Find the sum
v
→
0
+
v
→
1
+
v
→
2
+
v
→
3
. b. Find the coordinate vector of
v
→
0
with respect to the basis
v
→
1
,
v
→
2
,
v
→
3
. c. Let T he the linear transformation with
T
(
v
→
0
)
=
v
→
3
,
T
(
v
→
3
)
=
v
→
1
, and
T
(
v
→
1
)
=
v
→
0
. What is
T
(
v
→
2
)
? Describe the transformation T geometrically (as a reflection, rotation. projection, or whatever). Find the matrix B of T with respect to the basis
v
→
1
,
v
→
2
,
v
→
3
. What is
B
3
? Explain.
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.
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