Dr. Zs Math251 Handout (2nd ed.) #12.1 [Vectors in the Plane]
By Doron Zeilberger
Problem Type 12.1a: A person walks due west (or east) on the deck of a ship at a mi/h. This
ship is moving north (or south, or whatever) at b mi/h. Find the speed and direct
Dr. Zs Math251 Handout #13.3 (2nd ed.) [Arc Length and Speed]
By Doron Zeilberger
Problem Type 13.3a: Find the length of the curve r(t) = x(t)i + y (t)j + z (t)k , t0 t t1 .
Example Problem 13.3a: Find the length of the curve r(t) = t2 i + 2t j + ln t k ,
Dr. Zs Math251 Handout #13.4 (2nd ed.) [Curvature]
By Doron Zeilberger
Problem Type 13.4a: Find the curvature for
r(t) = x(t) i + y (t) j + z (t) k .
Example Problem 13.4a: Find the curvature for
r(t) = t i + 2t j + t2 k .
Steps
Example
1. Compute r (t) a
Dr. Zs Math251 Handout #13.2 (2nd ed.) [Calculus of Vector-Valued Functions]
By Doron Zeilberger
Problem Type 13.2a: Find a parametric equation for the tangent line to the curve with the
given parametric equation at the specied point
x = f1 (t) , y = f2 (
Dr. Zs Math251 Handout #13.1 (2nd ed.) [Vector-Valued Functions]
By Doron Zeilberger
Problem Type 13.1a: Find a vector equation and a parametric equation for the line segment
joining P (p1 , p2 , p3 ) and Q(q1 , q2 , q3 )
Example Problem 13.1a: Find a vec
Dr. Zs Math251 Handout #12.5 (2nd ed.) [Planes in Three Space]
By Doron Zeilberger
Problem Type 12.5a: Find an equation of the plane that passes through three given points
Example Problem 12.5a: Find an equation of the plane that passes through the points
Dr. Zs Math251 Handout #12.4 (2nd ed.) [The Cross Product]
By Doron Zeilberger
Problem Type 12.4a: Find the cross product a b and verify that it is orthogonal to both a
and b.
a = a1 , a2 , a3
,
b = b1 , b 2 , b 3
.
Example Problem 12.4a: Find the cross p
Dr. Zs Math251 Handout (2nd ed.) #12.2 [Vectors in three dimensions]
By Doron Zeilberger
Problem Type 12.2a: Show that the triangle with vertices P = (p1 , p2 , p3 ), Q = (q1 , q2 , q3 ),
R = (r1 , r2 , r3 ) is an equilateral triangle.
Example Problem 12.
Dr. Zs Math251 Handout (2nd ed.) #12.3 [Dot Product and the Angle Between Two Vectors]
By Doron Zeilberger
Problem Type 12.3a : Use vectors to decide whether the triangle with P (p1 , p2 , p3 ), Q(q1 , q2 , q3 ),
R(r1 , r2 , r3 ), is right-angled.
Example
Dr. Zs Math251 Handout #13.5 (2nd ed.) [Motion in Three-Space]
By Doron Zeilberger
Problem Type 13.5a: Find the velocity, acceleration, and speed of a particle with the given
position function.
r(t) = x(t) i + y (t) j + z (t) k .
Example Problem 13.5a: Fi