MST124 BOOK B UNIT 5
- Created by: Scar.Rose.1
- Created on: 05-06-22 14:03
1
Distance formula
The distance between the points (x_{1},y_{1}) and (x_{2},y_{2}) is:
sqrt((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2})
Midpoint formula
The midpoint of the line segment joining the points (x_{1}y_{1}) and (x_{2},y_{2}) is
((x_{1}+x_{2})/2) , ((y_{1}+y_{2})/2))
- if two lines are perpendicular, and not parallel to the axes, then the product of their gradients is -1
- if A and B are points that do not lie on the same horizontal or vertical line, then the gradient of the perpendicular bisector of AB is: -1/gradient of AB
To find the perpendicular bisector of a line segment joining two points (a,b) and (c,d):
- the points (x,y) on this bisector are exactly those that are equidistant from (a,b) and (c,d); that is those satisfying the equation sqrt((x-a)^{2} + (y-b)^{2}) = sqrt((x-c)^{2} + (y-d)^{2})
- so this is the equation of the perpendicular bisector
2
Standard form of the equation of a circle
The circle with centre (a,b) and radius r has equation
(x - a)^{2} + (y - b)^{2} = r^{2}
- to find points that lie on a given circle, substitute any particular value of x into the equation of a circle, and solve the resulting equation to find the corresponding values of y
- the equation you have to solve is quadratic, so you obtain two, one, or no values of y
- each of these gives a point on the circle
- you can also sub in a value of y to find a corresponding x value to find (possible) points on a circle
Strategy: To find the equation of the circle passing through three points:
- find the equation of the perpendicular bisector of the line segment that joins any pair of the three points
- find the equation of the perpendicular bisector of the line segment that joins a different pair of the three points
- find the point of intersection of these two lines (the centre of the circle) and the radius (distance from the centre to any of the three points)
3
- a line and a circle can have two, one, or no points of intersection
- to find the points, solve the equation of the line and the equation of the circle simultaneously
- use similar methods to find the points of intersection of a line and a parabola
- two circles may interesect at two points, one point or not at all
- to find the points of intersection, solve their equations simultaneously
- eliminate the terms in x^{2} and y^{2} by subtracting one equation from the other
- to specify the positions of points in three-dimensional space, we extend the usual two-dimensional Cartesian coordinate system, by introducing a third axis, the z-axis
- this axis is at right angles to the x- and y-axes
- the two different versions of a three-dimensional coordinate system are known as the right-handed coordinate system and the left-handed coordinate system
- the way of remembering which is which is to use the right-hand rule (figure 22, page 129 book B) and the left-hand rule
- another way to remember is the right-hand grip rule (page 130 book B)
4
Distance formula (three dimensions)
The distance between two points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is
sqrt((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2})
The standard form of the equation of a sphere
The sphere with centre (a,b,c) and radius r has equation
(x - a)^{2} + (y - b)^{2} + (z - c)^{2} = r^{2}
Zero vector
The zero vector, denoted by 0, is the vector whose magnitude is 0 and has no direction
- to vectors are equal if they have the same magnitude and direction
Triangle law for vector addition
To find the sum of two vectors a and b, place the tail of b at the tip of a. Then a + b is the vector from the tail of a to the tip of b
5
Parallelogram law for vector addition
To find the sum of two vectors a and b, place their tails together, and complete the resulting figure to form a parallelogram. Then a + b is the vector formed by the diagonal of the parallelogram, starting from the point where the tails of a and b meet.
- adding the zero vector to any vector leaves it unchanged
a + 0 = a
Negative of a vector
The negative of a vector a, denoted by -a, is the vector with the same magnitude as a, but the opposite direction.
Vector subtraction
To subtract b from a, add -b to a.
a - b = a + (-b)
6
Scalar multiple of a vector
Suppose that a is a vector. Then, for any non-zero real number m, the scalar multiple ma of a is the vector:
- whose magnitude is the modulus of m times the magnitude of a
- that has the same direction as a if m is +ve, and the opposite direction is m is -ve
Also, 0a = 0
Properties of vector algebra
The following properties hold for all vectors a, b and c, and all scalars m and n
a + b = b + a
(a + b) + c = a + (b + c)
a + 0 = a
a + (-a) = 0
m(a + b) = ma + mb
(m + n)a = ma + na
m(na) = (mn)a
1a = a
7
- a bearing is an angle between 0^{o} and 360^{o}, measured clockwise in degrees from north to the direction of interest
Opposite, corresponding and alternate angles
Where two lines intersect:
- opposite angles are equal
Where a line intersects parallel lines:
- corresponding angles are equal
- alternate angles are equal
(page 151, Book B)
Watch the video tutorial for Example 13 on page 152 of Book B
8
Component form of a vector
If v = ai + bj, then the expression ai + bj is called the component form of v
- the scalars a and b are called the i-component and the j-component, respectively, of v
If v = ai + bj + ck, then the expression ai + bj + ck is called the component form of v
- the scalars a, b and c are called the i-component, j-component and k-component of v
- the i and j components of a two-dimensional vector are alternatively called the x-component and the y-component
- the k-component in a three-dimensional vector can be called the z-component
Alternative component form of a vector
The vector ai + bj can be written in brackets, with a above b
The vector ai + bj + ck can be written in brackets too, with a at the top, b in the middle, and c at the bottom
- a vector written in this form is called a column vector
- the first number in a column vector is its i-component, the second number is its j-component, and, in three dimensions, the third number is its k-component
9
Addition of two-dimensional vectors in component form
If a = a_{1}i + a_{2}j and b = b_{1}i + b_{2}j, then
a + b = (a_{1} + b_{1})i + (a_{2} + b_{2})j
Addition of three-dimensional vectors in component form
if a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k, then
a + b = (a_{1} + b_{1})i + (a_{2} + b_{2})j + (a_{3} + b_{3})k
Negative of a two-dimensional vector in component form
if b = b_{1}i + b_{2}j, then -b = -b_{1}i - b_{2}j
Negative of a three-dimensional vector in component form
if b = b_{1}i + b_{2}j + b_{3}k, then -b = -b_{1}i - b_{2}j - b_{3}k
10
Subtraction of two-dimensional vectors in component form
if a = a_{1}i + a_{2}j and b = b_{1}i + b_{2}j, then
a - b = (a_{1} - b_{1})i + (a_{2} - b_{2})j
Subtraction of three-dimensional vectors in component form
if a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k, then
a - b = (a_{1} - b_{1})i + (a_{2} - b_{2})j + (a_{3} - b_{3})k
Scalar multiplication of a two-dimensional vector in component form
If a = a_{1}i + a_{2}j and m is a scalar, then
ma = ma_{1}i + ma_{2}j
Scalar multiplication of a three-dimensional vector in component form
If a = a_{1}i + a_{2}j + a_{3}k and m is a scalar, then
ma = ma_{1}i + ma_{2}j + ma_{3}k
11
- if P is any point, either in the plane or in three-dimensional space, then the position vector of P is the displacement vector OP, where O is the origin
- the components of the position vector of a point are the same as the coordinates of the point
- if the points A and B have position vectors a and b, respectively then the vector AB is equal to b - a
Midpoint formula in terms of position vectors
If the points A and B have position vectors a and b, respectively, then the midpoint of the line segment AB has position vector 1/2(a + b)
The magnitude of a two-dimensional vector in terms of its components
The magnitude of the vector ai + bj is sqrt(a^{2} + b^{2})
The magnitude of a three-dimensional vector in terms of its components
The magnitude of the vector ai + bj + ck is sqrt(a^{2} + b^{2} + c^{2})
12
Component form of a two-dimensional vector in terms of its magnitude and its angle with the positive direction
If the two-dimensional vector v makes the angle theta with the +ve x-direction, then
v = magnitude of v*cos(theta)i + magnitude of v*sin(theta)j
- the angle between two vectors a and b is the angle in the range 0 ≤ theta ≤ 180^{o} between their directions when the vectors are placed tail to tail
Scalar product of two vectors
The scalar product of the non-zero vectors a and b is
a.b = magnitude of a * magnitude of b * cos(theta)
where theta is the angle between a and b
- if two non-zero vectors are perpendicular, then their scalar product is zero
- so if the scalar product of two non-zero vectors is 0, then the vectors are perpendicular
- the scalar product of a vector with itself is equal to the square of the magnitude of the vector
13
Properties of the scalar product
The following properties hold for all vectors a, b and c, and every scalar m
- suppose that a and b are non-zero. If a and b are perpendicular, then a.b = 0, and vice versa
- a.a = (magnitude of a)^{2}
- a.b = b.a
- a.(b+c) = a.b + a.c
- (ma) . b = m(a.b) = a.(mb)
Scalar product of two-dimensional vectors in terms of components
If a = a_{1}i + a_{2}j and b = b_{1}i + b_{2}j, then
a.b = a_{1}b_{1} + a_{2}b_{2}
Scalar product of three-dimensional vectors in terms of components
if a = a_{1}i + a_{2}j + a_{3}k and b = b_{1}i + b_{2}j + b_{3}k then
a.b = a_{1}b_{1 }+ a_{2}b_{2} + a_{3}b_{3}
14
Angle between two vectors
The angle theta between any two non-zero vectors a and b is given by
cos(theta) = (a.b)/((magnitude of a)*(magnitude of b))
Related discussions on The Student Room
- A Level Mechanics. »
- Vectors »
- modelling in mechanics edexcel EX 8D »
- Edexcel IGCSE Higher tier Mathematics A Paper 1 1H (4MA1) - 19th May 2023 [Exam Chat] »
- Maths A-Level Vectors »
- can anyone answer this A level vectors question (very challenging) »
- Maths help - Vectors »
- Maths A Level Coordinate Geometry Question »
- Edexcel IGCSE Higher tier Mathematics A Paper 2 2H (4MA1) - 3rd June 2024 [Exam Chat] »
- Further math vecotr question »
Comments
No comments have yet been made