Orthogonal unit vectors in ℝ 3 Consider the vectors I = 〈 1 / 2 , 1 / 2 , 1 / 2 〉 , J = 〈 − 1 / 2 , 1 / 2 , 0 〉 , and K = 〈 1 / 2 , 1 / 2 , − 1 / 2 〉 . a. Sketch I , J , and K and show that they are unit vectors. b. Show that I , J , and K are pairwise orthogonal. c. Express the vector 〈1, 0, 0〉 in terms of I , J , and K .
Orthogonal unit vectors in ℝ 3 Consider the vectors I = 〈 1 / 2 , 1 / 2 , 1 / 2 〉 , J = 〈 − 1 / 2 , 1 / 2 , 0 〉 , and K = 〈 1 / 2 , 1 / 2 , − 1 / 2 〉 . a. Sketch I , J , and K and show that they are unit vectors. b. Show that I , J , and K are pairwise orthogonal. c. Express the vector 〈1, 0, 0〉 in terms of I , J , and K .
Solution Summary: The author illustrates how to sketch the vectors I, J, and K using an online graphing calculator.
Orthogonal unit vectors in
ℝ
3
Consider the vectors
I
=
〈
1
/
2
,
1
/
2
,
1
/
2
〉
,
J
=
〈
−
1
/
2
,
1
/
2
,
0
〉
, and
K
=
〈
1
/
2
,
1
/
2
,
−
1
/
2
〉
.
a. Sketch I, J, and K and show that they are unit vectors.
b. Show that I, J, and K are pairwise orthogonal.
c. Express the vector 〈1, 0, 0〉 in terms of I, J, and K.
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.
Let a be the angle between the vectors u=(3.0.4) and v-(4.0.3). Then*
O None of these
O sin(a)= 24/25
cos(a)= 24/25
cos(a)= - 24/25
Let v be a vector whose coordinates are given as v = [vx, Vy, Vz. If the
quaternion Q represents a rotation, show that the new, rotated coordinates of v are
given by Q(0, Vx, Vy, Vz)Q*, where (0, vx, Vy, Vz) is a quaternion with zero as its real
component.
Use a simple algorithm to do the following ant it as a matlab code
1. Determine the mean of the elements of a matrix A WITHOUT using the mean function
2. Find out how many non-zero elements a matrix A has
3.
Sum the diagonal elements of a square matrix A
4. Add the first and last elements of a vector v if you are not given the length of v
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