MAT 21C Lecture Notes - Lecture 17: Global Positioning System, Unit Circle, Dot Product
MAT 21C – Lecture 17 – Differentiation and Integration of Vectors
• Vector valued functions: If
, then
• Geometric interpretation:
is the position vector of point, p(t) in
the plane or space. is a tangent vector to the curve defined by
• Dynamical interpretation: is the position of a moving particle at time, t in a
plane or space. is the velocity of the particle.
• Example: Find an equation for the plane orthogonal to the curve
at point t = 1.
The normal vector to the plane is the tangent vector to the curve. Then
. At t = 1, and . The equation
for the plane is 1(x – 1) + 2(y – 1) + 3(z – 1) = 0 x + 2y + 3z = 6.
• Identity for derivative of the dot product: Let
and
Then
• Suppose is a vector whose length remains constant. Then
where is a constant. Differentiating, we have
. This implies that is
orthogonal to .
• Example: In the unit circle, if , then
representing the radius of 1. is orthogonal to .
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Mat 21c lecture 17 differentiation and integration of vectors: vector valued functions: if (cid:4666)(cid:1872)(cid:4667)= (cid:1858)(cid:4666)(cid:1872)(cid:4667)(cid:2835) +(cid:1859)(cid:4666)(cid:1872)(cid:4667)(cid:2836) + (cid:4666)(cid:1872)(cid:4667) , then (cid:4666)(cid:1872)(cid:4667)= Is the position vector of point, p(t) in: geometric interpretation: (cid:4666)(cid:1872)(cid:4667)=(cid:1841)(cid:1842)(cid:4666)(cid:1872)(cid:4667) the plane or space. (cid:4666)(cid:1872)(cid:4667) is a tangent vector to the curve defined by (cid:4666)(cid:1872)(cid:4667: dynamical interpretation: (cid:4666)(cid:1872)(cid:4667) is the position of a moving particle at time, t in a plane or space. (cid:4666)(cid:1872)(cid:4667) is the velocity of the particle: example: find an equation for the plane orthogonal to the curve (cid:4666)(cid:1872)(cid:4667)= < (cid:1872),(cid:1872)(cid:2870),(cid:1872)(cid:2871)> at point t = 1. The normal vector to the plane is the tangent vector to the curve. At t = 1, (cid:4666)(cid:883)(cid:4667)= and (cid:4666)(cid:883)(cid:4667)= . The equation for the plane is 1(x 1) + 2(y 1) + 3(z 1) = 0 (cid:3643) x + 2y + 3z = 6. Identity for derivative of the dot product: let (cid:1873) (cid:4666)(cid:1872)(cid:4667)= and (cid:1874) (cid:4666)(cid:1872)(cid:4667)= .