Vectors (P1.11 & 2.12)

Question 1

define vector

Answer 1

has both direction & magnitude

Question 2

define scalar

Answer 2

has magnitude only

Question 3

how can a vector be respresented?

Answer 3

directed line segment
column vector
i, j & k form

Question 4

PQ-> = RS->

Answer 4

line segments PQ & RS are equal in length & are parallel

Question 5

what defines parallel vectors?

Answer 5

scalar multiples of the same vector

Question 6

AB-> = -BA->

Answer 6

line segment AB is equal in length, parallel & in the opposite direction to BA

Question 7

what is the triangle law for vector addition?

Answer 7

AB-> + BC-> = AC->

Question 8

in Q, vertices of the given shape are labelled in order of the alphabet
e.g. A & B will be adjacent vertices

Question 9

how can a vector be described?

Answer 9

by its change in position or displacement relative to the x- & y- axes

Question 10

adding & multiplying column vectors

Question 11

describe a unit vector

Answer 11

magnitude/length of 1
unit vector along x-axis: i (1 0)
unit vector along y-axis: j (0 1)
â = a/|a|

Question 12

describe the magnitude-direction form of a vector

Answer 12

give the magnitude & angle b/w the vector & one of the axes

Question 13

what are the 5 techniques when solving geometric problems with vectors?

Answer 13

known ratio
unknown ratio
comparing coefficients
finding angles
parallel or co-linear
see Wilson OneNote Y1

Question 14

when solving geometric problems with vectors, how do you solve ‘if point P divides the line segment AB in the ratio λ:μ’?

Answer 14

see pg 244 P1 textbook

Question 15

equating coefficients in vector equations

Answer 15

a & b are non-parallel vectors
pa + qb = ra + sb
p=r & q=s

Question 16

give 3 examples of vector quantities in mechanics

Answer 16

velocity
displacement
force

Question 17

magnitude of a vector = scalar quantity
scalar of velocity vector
scalar of displacement vector

Answer 17

speed
distance

Question 18

what are position vectors?

Answer 18

location of point
measured from the origin

Question 19

AB-> =

Answer 19

AO-> + OB->

Question 20

what is the formula for the distance between points (x1, y1, z1) & (x2, y2, z2)?

Answer 20

√(x1-x2)^2 + (y1-y2)^2 + (z1-z2)^2

Question 21

what is the unit vector in the z direction?

Answer 21

k
(0 0 1)

Question 22

what is the formula for the angle vector a = xi + yj + zk makes with the positive x-axis?

Answer 22

cosθx = x / |a|

Question 23

what is the formula for the angle vector a = xi + yj + zk makes with the positive y-axis?

Answer 23

cosθy = y / |a|

Question 24

what is the formula for the angle vector a = xi + yj + zk makes with the positive z-axis?

Answer 24

cosθz = z / |a|

Question 25

solving 3d geometric problems involving vectors

Answer 25

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

Question 26

tools to help solve geometric problems involving vectors

Answer 26

1. sketching - ‘see’ the problem
2. notation:
   a. shape ABCD (clockwise or anti-clockwise around the shape - vertex to vertex)
   b. line segment AB vs. AB->
3. parallel vectors (a & λa where λ is a scalar)
4. basic geometry (area of shapes, properties of triangles & quadrilaterals)
5. find something else about the problem, anything!
   a. magnitudes of vectors
   b. vectors between points

Question 27

prove lines meet at a point & bisect each other

Answer 27

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->