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20 Cards in this Set

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What is the magnitude of a vector?
Eg. ||V|| = ?
sqrt(sum(each component of V squared))
What are the other names for the magnitude of a vector?
The norm or length
What is a unit vector?
A vector having magnitude of unit length (1).
How do you resize a vector V to unit length?

What is the requirement for doing this?
Divide each component by ||V||

At least one nonzero component
What is a normal vector?
A vector that is perpendicular to a surface at a particular point.
What is the scalar or inner product?
The dot product of two vectors
What does the dot product of two vectors mean?
It is a measure of the difference between the directions in which the two vectors point.
How do you find the dot product of two vectors?
It is the sum of the products of each of the corresponding components.

P dot Q = P1*Q1 + P2*Q2 + P3*Q3
If P and Q are perpendicular, what is their dot product?
0
How is the dot product related to the angle between two vectors?
P dot Q = ||P||*||Q|| cos (angle)
Will the dot product be positive or negative if the angle between P and Q is greater than 90?
The dot product will be negative.
What is the formula for the projection of P onto Q
(P dot Q)*Q/||Q||^2
What is the formula for the perpendicular component of P wrst Q?
P-(P dot Q)*Q/||Q||^2
What is the formula for P X Q, if P and Q are 3d vectors?
P X Q = <PyQz - PzQy, PzQx - PxQz, PxQy - PyQx>
How does ||P X Q|| relate to the angle between two vectors?
||P X Q|| = ||P||*||Q|| sin(angle)
How do you find the orthogonal vectors from n linearly independent vectors?
For each vector subtract the projection of that vector onto the other vectors from that vector.
What is the unit quaternion corresponding to a rotation through an angle theta about the unit axis A?
q = cos(theta/2)+A*sin(theta/2)
How are two quaternions spherically interpolated?
q(t) = q1(1-t)+q2(t)
How is a quaternion q applied to a point P?
P1 = qPq^-1
How should you transform a normal vector?
It is transformed using the inverse transpose of the matrix.