Vectors (P1.11 & 2.12) Flashcards
define vector
has both direction & magnitude
define scalar
has magnitude only
how can a vector be respresented?
directed line segment
column vector
i, j & k form
PQ-> = RS->
line segments PQ & RS are equal in length & are parallel
what defines parallel vectors?
scalar multiples of the same vector
AB-> = -BA->
line segment AB is equal in length, parallel & in the opposite direction to BA
what is the triangle law for vector addition?
AB-> + BC-> = AC->
in Q, vertices of the given shape are labelled in order of the alphabet
e.g. A & B will be adjacent vertices
how can a vector be described?
by its change in position or displacement relative to the x- & y- axes
adding & multiplying column vectors
describe a unit vector
magnitude/length of 1
unit vector along x-axis: i (1 0)
unit vector along y-axis: j (0 1)
â = a/|a|
describe the magnitude-direction form of a vector
give the magnitude & angle b/w the vector & one of the axes
what are the 5 techniques when solving geometric problems with vectors?
known ratio
unknown ratio
comparing coefficients
finding angles
parallel or co-linear
see Wilson OneNote Y1
when solving geometric problems with vectors, how do you solve ‘if point P divides the line segment AB in the ratio λ:μ’?
see pg 244 P1 textbook
equating coefficients in vector equations
a & b are non-parallel vectors
pa + qb = ra + sb
p=r & q=s
give 3 examples of vector quantities in mechanics
velocity
displacement
force
magnitude of a vector = scalar quantity
scalar of velocity vector
scalar of displacement vector
speed
distance
what are position vectors?
location of point
measured from the origin
position vector AB-> =
AO-> + OB->
what is the formula for the distance between points (x1, y1, z1) & (x2, y2, z2)?
√(x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2
what is the unit vector in the z direction?
k
(0 0 1)
what is the formula for the angle vector a = xi + yj + zk makes with the positive x-axis?
cosθx = x / |a|
what is the formula for the angle vector a = xi + yj + zk makes with the positive y-axis?
cosθy = y / |a|
what is the formula for the angle vector a = xi + yj + zk makes with the positive z-axis?
cosθz = z / |a|
solving 3d geometric problems involving vectors
use 2 routes to get to a given point
keep in vertex notation for as long as possible
if a, b & c are vectors in 3d that do not lie on the same plane (non-coplanar), then you can compare coefficients on both sides of an equation
simultaneous equations for coefficients
tools to help solve geometric problems involving vectors
- sketching - ‘see’ the problem
- notation:
a. shape ABCD (clockwise or anti-clockwise around the shape - vertex to vertex)
b. line segment AB vs. AB-> - parallel vectors (a & λa where λ is a scalar)
- basic geometry (area of shapes, properties of triangles & quadrilaterals)
- find something else about the problem, anything!
a. magnitudes of vectors
b. vectors between points
prove lines meet at a point & bisect each other
if your equation has a solution, it means the lines meet at a point
if your equation has no solution, it means the lines do not meet at a point
λ & μ = 1/2 means the lines bisect each other
ratio of λ:μ shows ratio of each line segment
conclusion: e.g. OX-> = 1/2OA->
