Sum and Difference
What is the difference between ⟨4, 2⟩ - ⟨6, -9⟩?
⟨-2, 11⟩
Find cos(⍺) (angle to x-axis) of ⟨7, 1, 8⟩
7/√(114) ≈ 0.66
Find the dot product of
⟨4, 2⟩ • ⟨6, -9⟩
6
Find the unit vector of 3i + 4j
⅗i + ⅘j
What two real-life applications are Quaternions used for?
Animation and Engineering
Find the sum of ⟨10,6⟩ + ⟨4,-5⟩
⟨14, 1⟩
Find the direction angle of <0,4,3> where
theta is greater than zero and less than or equal to 2 pi
cos-1(0/25)=pi/2
find the magnitude of <2,10>
square root of 104
Find the exact distance between⟨4, 2,7⟩ and ⟨6, -9,14⟩
square root of 174
Who created Quaternions and why?
William Rowan Hamilton because he thought calculus was inconvenient and geometrically unsatisfying
⟨6, 12⟩ - ⟨20, -9⟩+⟨4, -2⟩
⟨-10, 19⟩
Find angle between ⟨-4, 9⟩, ⟨7, -1⟩, exact or rounded to 2 decimal places
cos-1(-37√(197)/970) = 122.37 degrees
Find the dot product of
⟨3,10⟩ • ⟨20, -5⟩
10
What are the sum of the squares of the direction cosines equal to? Write the formula
cos2A+cos2B+cos2C=1
What do Quaternions do in animation?
Easier rotation of characters about multiple axes
3⟨2, 2⟩ +2⟨6, 4⟩ -5⟨3, 1⟩
⟨3, 9⟩
Find the vector orthogonal to ⟨4, 2, 0⟩ and ⟨3, 5, 7⟩
⟨14, -28, 14⟩
Find the exact magnitude of ⟨6, 2,7⟩
square root of 89
Write a vector in terms of magnitude and direction
V=||V||[(cos(A))i +(cos(B))i +(cos(C))i]
What do quaternions do in engineering? What other application can be used to do the same thing and why isn't it used?
1. Used to calculate the physical body rotational positions of 3D models in space
2. matrices can also be used but quaternions are more efficient because they represent theory of rotations