Vector Geometry - University of Kansaspeople.ku.edu/~f554h686/link_teaching/Vector_Geometry... · 2019-02-09 · Vector Addition and Scalar Multiplication Vector Addition Vectors

Vector Geometry

Vector Basics - Sections 12.1 and 12.2

Dot and Cross Products - Sections 12.3 and 12.4

Lines and Planes - Sections 12.2 and 12.5

MATH 127 (12.1 - 12.5) Vector Geometry The University of Kansas 1 / 16

Vectors

A vector is a geometric quantity described by magnitude (length) and direction. Ascalar is a constant in R which has no direction and is magnitude only.

Example: Force, flux, pressure, velocity, and acceleration.

Geometric Representations of Vectors: a segment AB from A to B with anarrow beginning at A and ending at B.

A - Initial Point B - Terminal Point

Notation: ~v =−→AB or v= ~AB

Only the magnitude and direction arerelevant to a vector. It’s location isNOT relevant. Vectors are invariantunder translation.


Vector Addition and Scalar Multiplication

Vector Addition

Vectors ~u and ~v are summed through theParallelogram Law.

Scalar Multiplication

When a vector ~v and a scalar c aremultiplied, the direction remainsunchanged and the magnitude of ~v ismultiplied by c .


Cartesian Representation of Vectors

Components of a Vector ~vPosition the initial point of the vector ~v at the origin O of the Cartesiancoordinate system. The coordinates of the terminal point P of ~v will be thecomponents of ~v .

~v =−→OP = 〈a, b〉

In general, if ~v =−→AB where A(x1, y1)

and B(x2, y2) then

~v = 〈 x2 − x1 , y2 − y1 〉


Vector Addition Scalar Multiplication

If ~v = 〈a1, b1〉 and ~u = 〈a2, b2〉 and k is a scalar, then

(I) ~v + ~u = 〈a1 + a2, b1 + b2〉(II) k~v = 〈ka1, kb1〉

(III) ~v − ~u = 〈a1 − a2, b1 − b2〉 = − (~u − ~v)


Vector Magnitude

The magnitude (or length) of a vector ~v = 〈a, b〉 is

|~v | =√a2 + b2

If ~v =−→AB with A(x1, y1) and B = (x2, y2), then

|~v | =√

(x2 − x1)2 + (y2 − y1)2


Algebraic Properties and Special Vectors

The zero vector is ~0 = 〈0, 0〉. The zero vector is the only vector with magnitudezero.

The Standard Basis Vectors are~i = 〈1, 0〉 and ~j = 〈0, 1〉.Note: 〈a, b〉 = a~i + b~j

A unit vector is a vector of magnitude one.

Two nonzero vectors ~v and ~w are parallel if they are scalar multiples of eachother; there exists a scalar c such that ~v = c ~w .


R3 Cartesian Coordinate System

Representing geometric objects in space by variables of numbers so that algebraand calculus can be used to study the geometric objects.

Setup of a Cartesian Coordinate System:

The origin O - the reference point.

Three coordinate axes: x-axis, y -axis, and z-axis.

Three coordinate planes: xy -plane, yz-plane, and xz-plane.

Eight octants.


In addition to the vector addition and scalar multiplication, there are otheroperations among vectors. In this course we will discuss two types of vectorproducts:

(I) The Dot Product (II) The Cross Product

The dot product of two vectors ~v = 〈a1, b1, c1〉 and~u = 〈a2, b2, c2〉 is the number given by

~v · ~u = |~u| |~v | cos (θ)

where θ is the small angle between the vectors ~v and~u.

~v · ~u = a1a2 + b1b2 + c1c2

Two vectors ~v and ~u are orthogonal to each other, denoted ~v ⊥ ~u, if and only ifθ = π

2 if and only if ~v · ~u = 0.


The cross product of two vectors ~v and ~u, denoted ~v × ~u, isthe vector

~v × ~u = (|~v | |~u| sin (θ)) ~n

(i) θ is the angle between ~v and ~u,

(ii) ~n is the unit vector, so that ~n ⊥ ~v , ~n ⊥ ~u, and thevectors (~v , ~u, ~n) satisfies the right-hand rule.

If ~v and ~u are parallel, then ~v × ~u = ~0

(~v × ~u) ⊥ ~v and (~v × ~u) ⊥ ~u

~v × ~u = −~u × ~v

|~v × ~u| is the area of the parallelogram:


~i × ~j = ~k ~j × ~k = ~i ~k × ~i = ~j

The determinant of a 2× 2 matrix is calculated:∣∣∣∣ a bc d

∣∣∣∣ = ad − bc

The determinant of a 3× 3 matrix can be calculated by decomposing into a linearcombination of 2× 2 matrices.∣∣∣∣∣∣

a1 a2 a3b1 b2 b3c1 c2 c3

∣∣∣∣∣∣ = a1

∣∣∣∣ b2 b3c2 c3

∣∣∣∣ − a2

∣∣∣∣ b1 b3c1 c3

∣∣∣∣ + a3

∣∣∣∣ b1 b2c1 c2

∣∣∣∣~v × ~u = 〈v1, v2, v3〉 × 〈u1, u2, u3〉 =

∣∣∣∣∣∣~i ~j ~kv1 v2 v3u1 u2 u3

∣∣∣∣∣∣MATH 127 (12.1 - 12.5) Vector Geometry The University of Kansas 11 / 16

Lines in 2-Space

The slope of a line can be represented usinga directional vector.

The line y = −0.5x + 5 has slope m = −0.5;that is, when the x-value changes by +2, they -value changes by −1. The line moves inthe direction of the vector

~v = 〈2,−1〉

Any point Q on the line can be described by a vector function:

The vector−→PQ is parallel to ~v .

There is a scalar t where−→PQ = t~v .

Using the parallelogram law:

~r(t) = ~rp + t~v ~r(t) = 〈2, 4〉+ t〈2,−1〉


Lines in 3-Space

Recall the point-slope form of a line in a plane: Given a point (x0, y0) on a linewith slope (direction) m, the equation of the line can be written

y − y0 = m(x − x0)

In space, a line can still be determined by a point and a direction.

Let P0(x0, y0, z0) be a point and ~r0 = 〈x0, y0, z0〉 be the position vector of P0. Let~v = 〈v1, v2, v3〉 be a vector.

We’ll discuss three types of equations ofthe line L that passes through the pointP0 with the direction ~v .


Equations of the line L in 3-Space

The vector equation of L is

~r − ~r0 = t~v for all t

~r (t) = 〈x0 + tv1, y0 + tv2, z0 + tv3〉

The parametric equation of L is

x = x0 + tv1, y = y0 + tv2, z = z0 + tv3

The symmetric equation of L is

x − x0v1

=y − y0v2

=z − z0v3

provided that v1, v2, v3 6= 0. If, for instance, v1 = 0, then the equation is

x = x0 ,y − y0v2

=z − z0v3


Planes in Space

Let P0(x0, y0, z0) be a point and~r0 = 〈x0, y0, z0〉. Let ~n = 〈n1, n2, n3〉 bea nonzero vector. Then there is a planethat passes through P0 and is orthogonalto ~n.

If P(x , y , z) is a general point on the plane and ~r = 〈x , y , z〉 is the position vectorof P, then the vector equation of the plane is

~n · (~r − ~r0) = 0

The scalar equation of the plane is

n1(x − x0) + n2(y − y0) + n3(z − z0) = 0

The vector ~n = 〈n1, n2, n3〉 is a normal vector to the plane.


Relative Positions of Two Lines in SpaceTwo lines can be parallel. The direction vectors for the lines are scalarmultiples of one another. Identical lines are parallel.

Two non-parallel lines can intersect at a point.

Two lines can be skew. Skew lines are not parallel and do not intersect.

Relative Positions of Two Planes in SpaceTwo planes can be parallel. Their normal vectors are scalar multiples of oneanother. Identical planes are parallel.

Two planes can intersect.

When two planes intersect, their intersection is a line.The angle θ between is the acute angle between their normal vectors. If θ = 0then the planes are parallel.


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Vector Geometry - University of Kansaspeople.ku.edu/~f554h686/link_teaching/Vector_Geometry... · 2019-02-09 · Vector Addition and Scalar Multiplication Vector Addition Vectors