JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

MATH 151 Engineering Mathematics ISpring 2018, WEEK 1

JoungDong Kim

Week 1 Vectors, The Dot Product, Vector Functions and Parametric Curves.

Section 1.1 Vectors

Definition. A Vector is a quantity that has both magnitude and direction. A two-dimensionalvector is an ordered pair a = 〈a1, a2〉 of real numbers. The numbers a1 and a2 are called thecomponents of a. a1 is x component, a2 is y component.

Vector has an initial point and terminal point

Note. If the initial point is origin, we call position vector.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Given the points A(x1, y1) and B(x2, y2), the vector−→AB is 〈x2 − x1, y2 − y1〉

Ex.1) Find the vector presented by the directed line segment with initial point A(1, 2) and terminalpoint B(4,−3).

The length of the vector a = 〈a1, a2〉 is |a| =√

a21+ a2

2

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Vector Addition. If a = 〈a1, a2〉 and b = 〈b1, b2〉, then the vector a+ b is defined by

a+ b = 〈a1 + b1, a2 + b2〉

Multiplication of a vector by a scalar. If c is a scalar and a = 〈a1, a2〉, then the vectorca is defined by

ca = 〈ca1, ca2〉

Ex.2) If a = 〈4, 3〉 and b = 〈−2, 1〉, find |a| and the vectors a+ b, a− b, 3b, and 2a+ 5b.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

If v has direction angle θ, the components of v can be computed using the formula

v = 〈|v| cos θ, |v| sin θ〉

Ex.3) Find the component of the vector r given that;

a) |r| = 2 and r makes an angle of 60◦ with the positive x-axis.

b) |r| = 7 and r makes an angle of 150◦ with the positive x-axis.

c) |r| =1

2and r makes an angle of −45◦ with the positive x-axis.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Basis Vector (Standard) i = 〈1, 0〉 , j = 〈0, 1〉

Unit Vector is a vector with length one,a

|a|.

Ex.4) If a = 〈−2, 1〉, find a unit vector of a and a vector with length 3 in the direction of a.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Applications to Physics and Engineering

A force is represented by a vector because it has both a magnitude and direction. Ifseveral forces are acting on an object, Resultant Force experienced by the object isthe vector sum of the forces.

Ex.5) John walks due west on the deck of a ship at 3mph. The ship is moving north at 22mph. Findthe speed and direction of John relative to the surface of the water.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.6) Two forces F1 and F2 with magnitudes 10 lb and 12 lb act on an object at a point P as shown inthe figure. Find the resultant force F acting at P as well as its magnitude and its direction.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.7) Suppose that a wind is blowing into the direction S45◦E at a speed of 50 km/h. A pilot is steeringa plane in the direction N60◦E at an speed of 250 km/h. Find the true course (direction of theresultant velocity vector of the plane and wind) and ground speed(magnitude of resultant).

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Section 1.2 The Dot Product

The work done by a constant force F in moving an object through a distant d is

W = Fd,

but this applies only when the force is directed along the line on motion of the object.

Suppose the constant force is a vector ~F =−→PR pointing in some other dirction. If the force moves

the object from P to Q, then the displacement vector is ~D =−→PQ. The magnitude of the force

applied in the direction of motion, that is

|−→PS| = |~F | cos θ

Thus the work done W in moving the object is given by

W = |~F || ~D| cos θ.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

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.

Ex.8) If the vectors a and b have lengths 4 and 6, and the angle between them isπ

3, find a · b.

Ex.9) Find the work done by a force of 20 lbs acting in the direction N50◦W in moving an object 4 feetdue west.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.10) a is perpendicular to b, a · b is?

Two vectors a and b are orthogonal if and only if a · b = 0.

Ex.11) a and b are parallel, a · b is?

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

When we know the components of vectors, the dot product of a = 〈a1, a2〉 andb = 〈b1, b2〉 is

a · b = a1b1 + a2b2

Ex.12) Dot product of a = 〈2, 4〉 and b = 〈3,−1〉.

Ex.13) Find the values of x for which the vector 〈x, 5x〉 and 〈x,−10〉 are perpendicular.

Ex.14) Find the values of x for which the vector 〈2, x〉 and 〈x− 1, 3〉 are parallel.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.15) Find the angle between the vectors a = 〈2, 2〉 and b = 〈5,−3〉.

Ex.16) Find the angle between the vector 〈1, 5〉 and 〈−2, 3〉.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.17) 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 magnitudeof the force is measured in Newtons.

Ex.18) A woman exerts a horizontal force of 65lb on a crate as she pushes it up a ramp that is 20ft longand inclined at an angle of 20◦ above the horizontal. Find the work done on the box.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Projections

If S is the foot of the perpendicular from R to the line containing−→PQ, then the vector with

representation−→PS is called the Vector Projection of b onto a and is denoted by projab.

The Scalar Projcetion of b onto a. (also called the component of b along a) is defined tobe the magnitude of the vector projection, which is the number |b| cos θ, where θ is the anglebetween a and b, denoted by compab.

• The Scalar Projection of b onto a:

compab =a · b

|a|

• The Vector Projection of b onto a:

projab =

(

a · b

|a|

)

a

|a|=

a · b

|a|2· a

Ex.19) Find the vector and scalar projection of 〈4, 8〉 onto 〈2, 1〉.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Definition: Given the nonzero vector a = 〈a1, a2〉, the orthogonal complement of a is thevector a⊥ = 〈−a2, a1〉.

Ex.20) Find the orthogonal complement of a = 〈−1, 4〉. Graph both a and a⊥ on the same axis.

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Ex.21) Find two unit vectors perpendicular to 〈2,−3〉.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.22) Find the distance from the point P (2, 1) to the line y = 2x+ 1.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

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 itsdomain, we get a collection of points (x, y) = (f(t), g(t)) which traces out the parametric curves.

Ex.23) 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. And also we couldcombine these parametric functions as Cartesian equation.

Ex.24) Let x = 1− 2t, y = 2 + 3t, − 3 ≤ t < 3, find the Cartesian equation.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.25) x = t + 1, y = t2 − 4.

Ex.26) x = 2 sin θ, y = 3 cos θ.

Ex.27) x = sin t, y = csc t,π

6≤ t <

π

2.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Vector Function

We call r(t) = 〈x(t), y(t)〉 a Vector Function.

Ex.28) Sketch the following curves described by the vector function. Include the direction of the curveas t increases.

a) r(t) = 〈t− 1, 2− 3t〉.

b) r(t) = 〈2 + cos t, 1 + sin t〉.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Vector Equation of Line

A vector equation of the Line passing through the point r0 = (x0, y0) and parallel to the vectorv = 〈v1, v2〉 is given by

r(t) = r0 + tv

From this vector equation, we can obtain the parametric equations of the line as follows;

r(t) = r0 + tv

= 〈x0, y0〉+ t 〈v1, v2〉

= 〈x0 + tv1, y0 + tv2〉

Parametric equations of the line that passes through the point (x0, y0) and is parallel to thevector 〈v1, v2〉 are given by

x(t) = x0 + tv1, y(t) = y0 + tv2

Ex.29) Find the vector equation of the line parallel to the vector 〈1, 4〉 and passing through the point(−1, 5).

Note. If v is parallel to the line, then any multiple of v is also parallel to the line and can beused to obtain a vector equation or parametrized equations of a line.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.30) Find parametric equations for the line with slope4

3and passing through the point (2,−5).

Ex.31) Find the vector equation of the line passing through the points (1, 2) and (−1, 4).

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.32) Consider the line 2x+ 3y = 5.

a) Find a vector parallel to the line.

b) Find a vector perpendicular to the line.

Ex.33) An object is moving in the xy-plane and its position after t seconds is given by r(t) = 〈t+ 4, t2 + 2〉,

a) Find the position of the object at t = 2.

b) At what time does the object reach the point (7, 11)?

c) Eliminate the parameter to obtain a cartesian equation.

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JD Kim Texas A&M University Math 151 Engineering Math I, Spring 2018, Week 1

Ex.34) Consider the lines r(t) = 〈−4 + 2t, 5 + t〉 and s(w) = 〈2 + 3w, 4− 6w〉. Determine whether thelines are parallel, perpendicular or neither. If they are not parallel, find the intersection point.

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