Vectors

how do you find the equation of a line in 3d with position vector a and a parallel vector (scalar) b

a+λb

r=a+λb

Equation of a straight line

r - position vector of general point on the line

a - position vector of fixed point on the line

λ - scalar that varies with the point

b - direction vector (parallel to the line)

how do you find the vector equation of a line in 3d with position vectors a and b, both passing through a and b

r=a+λ(b-a)

How to find the point of intersection of two lines

1. Put the equations of the lines equal to each other

2. Solve two of the equations simultaneously

3. Substitute the values for the scalars (μ&λ) into the third equation, if it works then the lines intersect

4. Substitute one of the scalars into the equation of its line to find point of intersection

how to you write the equation of a line in 3d in cartesian form a=(a1,a2,a3) b=(b1,b2,b3) r=a+λb

(x-a1)/b1 = (y-a2)/b2 = (z-a3)/b3

the ACUTE angle between two straight lines:

cosθ = | (a•b)/(|a||b|) |

the OBTUSE angle between two straight lines

180-cosθ = | (a•b)/(|a||b|) |

do two lines have to meet to find their angle

no, you can imagine moving them to the origin(taking out their position vectors) and looking at the directions, find the angle they make

do you use the position vector or the direction vector to find the angles between two lines?

direction vector

how do you find the angle between two planes

you use their normals as the angle between the normals is the same at the angle between them. so:

cosθ = | (n1•n2)/(|n1||n2|) |

what does it mean if two vectors have a scalar product of 0?

they are perpendicular

what does scalar product mean?

Dot product of two lines direction vectors

how do you show the vector of a plane as a cartesian equation

n1x+n2y+n3z+d=0

n = the normal to the plane

x,y,z are variables/direction vectors

d = -(a•n) where a is the position vector and n is normal to the plane

what is the scalar product form of a plane

r•n=k

r is a variable/direction vector

n is the normal to the plane

k is a•n where a is any position vector on the plane

equation of a plane in vector form

r = a+λb+μc

a is a position vector on the plane

b and c are non parallel non zero vectors in the plane

FREEBIE

draw graphs of planes etc to understand the questions better

how do you find the angle between a line and a plane?

cosφ=(b•n)/(|b|n|)

then θ = 90-φ

(since you found the angle between the normal and the line)

if φ is obtuse: 180-φ=ι then 90-ι = θ

what is the r in r = ?

x,y,z

how to find the angle between two vectors

Cosθ=a.b/|a||b|

1. Find the scalar products of the vectors

2. Find the magnitude of the vectors

3. Use the equation to find the angle

Finding the angle between two lines

Use the same formula as with vectors, only use the direction vectors of the lines

r=a+λb+μc

r - position vector of a general point on the plane

a - position vector of a fixed point on the plane

λ&μ - scalars that vary with the point

b&c - non-parallel vectors that lie in the plane

How to find the equation of a plane when given 3 points (P,Q,R)

Use point P for a

Use PQ and PR for b and c

r.n=a.n=p

r - position vector of a general point on the plane

n - a normal to the plane

a - position vector of a fixed point on the plane

p - scalar constant

How to derive the Cartesian form of a planar equation from the scalar form

For plane r.n = p, if n =(a b c) and p= -d, then for any point (x,y,z) on the plane: (x y z).(a b c) = -d

Therefore ax+by+cz= -d

Therefore ax+by+cz+d=0

How to find the angle between a plane and a line

1. Find the angle between the line and the normal to the plane using scalar product

2. Add 90° (??)

How to find the angle between two planes

Use the scalar product to find the angle between the normals

Subtract this from 180°

How to find the point of intersection of a plane and a line

1. Write the equation of the

line as a single vector.

2. Substitute this vector into

the equation of the plane.

3. Solve to find λ.

4. Substitute the value of

λ into the equation of

the line-this gives the

point of intersection.

How to find the line of intersection of two planes

1. Write the equation of each plane in Cartesian form.

2. Use simultaneous equations to eliminate one variable so that you can express one variable in terms of another.

3. Substitute this back into the equation of one of the planes to get an equation in terms of the eliminated variable.

4. Change the subject of the equation you found in step 2 this gives you two different equations for one variable.

5. Write these as a single equation set to equal λ to give you a Cartesian equation for the line of intersection.

How to find the perpendicular distance between a point and a line

1. Find an equation for the line

i. Use the position vector of either point for

a

ii. The direction vector is the vector between the two points on the line

2. Find a>b when b is a general point on line l1

3. If a>b is perpendicular to the line, the scalar product of the direction vectors is 0, from this find λ

4. Then substitute in and find the distance a>b using Pythagoras

How to find the perpendicular distance between a point and a plane

Use the equation! 😊😆

(It's in the formula book)

How to find the perpendicular distance between parallel lines

1. Find A>B when A is a point on line l1 and B is a point on l2 by doing one line equation minus the other

2. Do that thing where λ-μ=t

3. Do the perpendicular business, scalar product = 0, to find t

4. Use the value of t to find the length of A>B

How to find the shortest distance between skew lines

(c-a).(bxd)/IbxdI

how to find the angle between a plane and a vector

sin theta =| |b⋅a| / |b| |n| |

What does it mean if 2 lines are skew

Lines are not parallel nor do they intersect