Vectors
Introduction to Vectors
scalar: a quantity that has a magnitude (size) but no direction
vector: has magnitude and a direction
tail: the point where a vector begins
head: the point where a vector ends
If the tail of the vector is at the origin and the head is at a point not located at the origin, the vector is defined as a position vector
The magnitude or modulus of the vector is the distance between the tail and head
The magnitude of the vector can never be negative, only positive or zero
If z is a complex number, then the absolute value of z represents the magnitude as the distance from (0,0) in the complex plane
A zero vector has a magnitude of zero and has no direction
Two vectors are said to be equal if they have the same magnitude and direction
The negative of a vector has the same magnitude but opposite direction
Every vector has two component vectors, one that is horizontal (starting at the tail of the vector) and one that is vertical (ending at the head of the original vector)
To find the component vectors for a defined vector, first subtract the degrees given from 90 and then multiply by the magnitude. The horizontal vector will use the cos of the angle, and the vertical will use sin. Factor in what quadrant the vector is in
In general, for any scalar k, the scalar product ka is a vector whose magnitude is k times that of a. Ka is parallel to a. If k > 0, then ka and a have the same direction. If k < 0, they have opposite directions, and if k = 0, then ka is a zero vector
Unit Vectors 2D
a unit vector is any vector with a magnitude of 1
In general, a unit vector, called a hat in the same direction as vector a is defined as a hat = 1/ the magnitude of a times a
To get a unit vector, multiply the vector by the reciprocal of its magnitude
A base vector is a horizontal unit vector j and a vertical unit vector i
Scalar Products
the scalar product or dot product of two vectors, a and b, is defined as a dot b = magnitude of a x magnitude of b x cos0
0 is the angle between a and b
the scalar product produces a scalar and not a vector
the scalar product is also equal to the sum of the components where x1 x x2 plus y1 x y2 = a dot b
if a and b are perpendicular, then a dot b = 0
for any vector a, a dot 0 = 0
if a and b are in the same direction, then a dot b = magnitude of a x magnitude of b
if a and b are in the opposite direction, then a dot b = - magnitude of a x magnitude of b
Direction Cosines
if a vector is “unitized” then the components of the vector are the direction cosines of that vector to the positive axis \
r hat = cosa i + cosb j + cosy k
cosa1 x cosa2 + cosb1 x cosb2 + cosy1 + cosy2 = cos 0
2-D Lines
given r = a + b
goal is to represent every point on the line
in order to do that, using parameter (scalar) t allows b to vary in length yet along the same (or opposite) direction as the line
with this change, r changes position relative to t
changes equation to r = a + tb
a is the position vector who defines any single point on the line
b is the direction who defines the slope of the lineb relative to the two axes
basis vector form is r = (x1i + y1j) + t (x2i + y2j) or (x1 + x2t)i + (y1 + ty2)j
column vector form is (x1 + tx2)/(y1 + ty2)
can also be written as x = x1 + x2t and y = y1 + y2t
to write the equation of the line in Cartesian form, y - y1 = y2/x2 (x - x1)
three possibilties for line intersection
1. they intersect at one point
2. they are parallel and never intersect
3. they are conincident and intersect at every point (also parallel)
lines are parallel if their direction vectors are scalar multiples of one another’
3-D Lines
still in form r = a + tb with the addition of point z
r = (x1i + y1j + z1k) + t (x2i + y2j + z2k)
two lines in three dimensions have four possibilites of intersecting
1. they intersect at a single point (co-planar)
2. they are parallel and don’t intersect (co-planar)
3. they are coincident (co-planar)
4. they are skew to each other (non-intersecting, non-parallel and not co-planar)
Vector Product
the vector product (cross product) is a cross b = magnitude of a x magnitude of b x sin 0 x n hat
0 is the angle between a and b
n hat is the unit vector perpendicular to a and b
the vector product produces a vector perpendicular to a and b (within the plane containing a and b) and has a magnitude of magnitude of a x magnitude of b x sin 0
a cross b = -b cross a
vector product is not communative
wheel is i to j to k (positive)
the volume of a parallelepid is formed by a cross b dot c
Planes
a plane is defined by three points
a position vector and two direction vectors
if a and b a vectors in a plane containing points A and B, then a cross b is a vector perpendicular to the plane (normal)
this means vector AP dot n = 0
produces rectangular equation a(x - x1) + b(y - y2) + c(z - z2) = 0
simplfies to ax + by + cz = D
two non-parallel planes must intersect in a line
if planes are in rectangular form as a1x + b1y + c1z = d1 and a2x + b2y + c2z = d2
the planes will be parallel if a1, b1, c1 = t(a2, b2, c2)
they will concident if that is true and d1=td2
if they are not scalar multiples, then they are intersecting
perpendicular if a1, b1, c1 dot (a2, b2, c2) = 0
Three planes in a space can have no common points by being all parallel, two are parallel and the third is intersecting (two parallel lines of intersection), two are coincident and one is parallel, all three are non-parallel, but each intersects the two in a line (three parallel lines of intersection)
three planes in a space can have one common points by all three being non-parallel and intersecting at one point
three planes in a space can have infinite points in common by two being conincident and the third intersecting these two, all three are non-parallel and intersect, like pages of a book, or all three being conicident
to find intersection use rref
if it produces a bottom row of 0001, then there are no common points
if it produces 0000, then it has infinitely many common points