9. Vectors Flashcards
What is r = a + λb?
Vector equation of straight line passing through point A with position vector a and parallel to vector b where λ is a scalar parameter
What is r = c + λ(d - c)?
Vector equation of straight line passing through points C and D with position vectors c and d respectively where λ is a scalar parameter
a = (a1 a2 a3)
b = (b1 b2 b3)
Equation of line with vector equation is r = a + λb
What is the Cartesian form of the equation of the line?
(x - a1) / b1 = (x - a2) / b2 = (x - a3) / b3
where each of these expressions is equal to λ
What is r = a + λb + μc?
Vector equation of a plane
r is a position vector of a general point in the plane
a is the position vector of a point in the plane
b and c are non-parallel, non-zero vectors in the plane
λ and μ are scalars
What is the Cartesian equation of a plane in 3 dimensions?
ax + by + cz = d
a, b, c and d are constants
(a b c) is normal vector to plane
What is the scalar product of two vectors a and b?
a. b (“a dot b”)
a. b = |a| |b| cos θ is the definition of the scalar (dot) product
a and b must be facing away from a point
θ is the angle between a and b
If a and b are position vectors of points A and B how do you work out the angle between them?
cos (AOB) = (a.b) / |a| |b|
When are the non-zero vectors a and b perpendicular?
a.b = 0
What is true about the dot product of a and b if they are parallel?
a.b = |a| |b|
so a.a = |a| ^ 2
a = a1 i + a2 j + a3 k b = b1 i + b2 j + b3 k
What is their scalar product?
a.b = (a1 a2 a3) . (b1 b2 b3)
= a1 b1 + a2 b2 + a3 b3
What equation is the acute angle between 2 intersecting straight lines given by?
cos θ = |(a.b) / (|a| |b|)|
a and b are direction vectors of lines
What is the scalar product form of the equation of a plane?
r.n = k
where k = a.n for any point in the plane with position vector a and n is the direction vector normal to the plane
What formula gives the acute angle θ between the line r = a + λb and the plane with equation r.n = k?
sin θ = |(b.n) / (|b| |n|)|
What formula gives the acute angle between the plane with equation r.n1 = k1 and the plane with equation r.n2 = k2?
cos θ = |(n1.n2) / (|n1| |n2|)|
What are the two characteristics of skew lines?
- They are not parallel
2. They do not intersect
What is the unique line segment AB where A lies on l1 and B lies on l2 and l1 and l2 are non-intersecting
For any two non-intersecting lines l1 and l2 there is a unique line segment AB such that A lies on l1 and B lies on l2 and AB is perpendicular to both lines
How is a perpendicular line drawn from point P to the line l?
A line drawn from P at right angles to the line l
How is a perpendicular line drawn from point P to a plane Π?
A line is drawn from P parallel to the normal vector n
What is shown in this equation r.n ̂ = k?
k is the length of the perpendicular from the origin to the plane Π where the equation of the plane is written in the form of r.n ̂ = k
n ̂ is a unit vector perpendicular to Π
What is shown in the equation |aα+bβ+cγ-d|/√(a^2+b^2+c^2)?
The perpendicular distance from the point with co-ordinates (α, β, γ) and the plane with equation ax + by + cz = d
What is meant by collinear?
If points all lie on the same straight line
What is meant by coplanar?
If points all lie on the same plane
What are the results for parallel unit vectors?
i. i = j.j = k.k = 1
i. j = i.k = j.i = j.k = k.i = k.j = 0
What do modulus signs ensure?
The angles are acute
