MATH 151 Engineering Math I, Spring 2014
JD Kim
Week2 Section 1.2, 1.3, 2.2, Dot product, parametrized curves, (qualitative) definition of limit
Section 1.2 The Dot Product
The work done by a constant force F in moving an object through a distant d is
W = F d, but this applies only when the force is directed along the line on motion
of the object.
Definition. The dot product of two nonzero vectors a and b is the number
a · b = |a||b| cos θ
where θ is the angle between a and b, 0 ≤ θ ≤ π. If either a or b is 0, we define
a · b = 0.
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Ex1) If the vectors a and b have lengths 4 and 6, and the angle between them is
π
, find a · b.
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Ex2) Find the work done by a force of 20lbs acting in the direction N50◦ W in
moving an object 4f eet due west.
When we know the components of vectors, the dot product of a =<
a1 , a2 > and b =< b1 , b2 > is
a · b = a1 b1 + a2 b2
Ex3) Dot product of a =< 2, 4 > and b =< 3, −1 >.
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Ex4) Find the angle between the vectors a =< 2, 2 > and b =< 5, −3 >.
Ex5) ~a is perpendicular to ~b, a · b is?
Thus if two vectors are perpendicular, dot product is zero and vise
versa.
Ex6) ~a and ~b are parallel, a · b is?
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Ex7) Find the values of x for which the vector < x, 5x > and < x, −10 > are
perpendicular.
Ex8) Find the values of x for which the vector < 2, x > and < x − 1, 3 > are
parallel.
Ex9) Find the angle between the vector < 1, 5 > and < −2, 3 >.
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Ex10) A force with representation F~ =< 3, 8 > moves an object along a straight
line from the point (2, 3) to the point (4, 5). Find the work done if the distance is
measured in meters and the magnitude of the force is measured in Newtons.
Ex11) A woman exerts a horizontal force of 65lb on a crate as she pushes it up
a ramp that is 20f t long and inclined at an angle of 20◦ above the horizontal. Find
the work done on the box.
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Projections
The Vector Projection of b onto a. If S is the foot of the perpendicular from
R to the line containing P~Q, then the vector with representation P~S is called the
Vector Projection of b onto a and is denoted by proja b.
a·b
a·b a
=
·a
proja b =
|a| |a|
|a|2
The Scalar Projcetion of b onto a. (also called the component of b along a)
is defined to be the magnitude of the vector projection, which is the number |b| cos θ,
where θ is the angle between a and b, denoted by compa b.
compa b =
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a·b
|a|
Ex12) Find the vector and scalar projection of < 4, 8 > onto < 2, 1 >.
Ex13) Find the vector projection of < 2, 1 > onto < −5, 1 >.
Ex14) Find the distance from the point P (2, 1) to the line y = 2x + 1.
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Definition: Given the nonzero vector a =< a1 , a2 >, the orthogonal
complement of a is the vector a⊥ =< −a2 , a1 >.
Ex15) Find the orthogonal complement of ~a =< −1, 4 >. Graph both ~a and ~a⊥
on the same axis.
Ex16) Find two unit vectors perpendicular to < 2, −3 >.
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Section 1.3 Vector Functions.
Parametric Curves
We call x = f (t) and y = g(t) parametric equations where t is the parameter.
As t varies over its domain, we get a collection of points (x, y) = (f (t), g(t)) which
traces out the parametric curves.
Ex17) Let x = t − 3, y = 2t − 1, Find x and y at t = 0, t = 1, t = 2.
Degree of t is one, means this parametric equation represents straight line.
Find the Cartesian equation of this.
Ex18) x = 1 − 2t, y = 2 + 3t, − 3 ≤ t < 3.
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Ex19) x = t + 1, y = t2 − 4.
Ex20) x =
√
t, y = 1 − t.
Ex21) x = 2 sin θ, y = 3 cos θ.
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Ex22) x = sin t, y = csc t,
π
π
≤t< .
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Vector Function
We call ~r(t) =< x(t), y(t) > a Vector Function.
Ex23) Sketch the following curves described by the vector function. Include the
direction of the curve as t increases.
Ex23-1) ~r(t) =< t − 1, 2 − 3t >.
Ex23-2) ~r(t) =< 2 + cos t, 1 + sin t >.
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Vector Equation of Line
A vector equation of the Line passing through the point r0 = (x0 , y0 ) and parallel
to the vector ~v =< v1 , v2 > is given by
~r(t) = r~0 + t~v
From this vector equation, we can obtain the parametric equations of the line as
follows;
~r(t) = r~0 + t~v
= < x0 , y0 > +t < v1 , v2 >
= < x0 + tv1 , y0 + tv2 >
(1)
(2)
(3)
Then x = x0 + tv1 and y = y0 + tv2 are parametric equations of the line.
Note that if v is parallel to the line, then any multiple of v is also parallel to
the line and can be used to obtain a vector equation or parametrized equations of a
line.
Ex24) Find the vector equation of the line parallel to the vector < 1, 4 > and
passing through the point (−1, 5).
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Ex25) Find parametric equations for the line with slope
the point (2, −5).
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and passing through
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Ex26) Find the vector equation of the line passing through the points (1, 2) and
(−1, 4).
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Ex27) Consider the line 2x + 3y = 5.
Ex27-1) Find a vector parallel to the line.
Ex27-2) Find a vector perpendicular to the line.
Ex28) An object is moving in the xy-plane and its position after t seconds is
given by ~r(t) =< t + 4, t2 + 2 >,
Ex28-1) Find the position of the object at t = 2.
Ex28-2) At what time does the object reach the point (7, 11)?
Ex28-3) Eliminate the parameter to obtain a cartesian equation.
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Ex29) Consider the lines ~r(t) =< −4 + 2t, 5 + t > and ~s(w) =< 2 + 3w, 4 − 6w >.
Determine whether the lines are parallel, perpendicular or neither. If they are not
parallel, find the intersection point.
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