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