vectors

Question 1

scalar product

Answer 1

multiply corresponding terms then add each of the products

Question 2

angle between two vectors. cosθ =

Answer 2

cosθ = a.b/(|a||b|)

Question 3

scalar product of perpendicular lines

Answer 3

a.b = 0

Question 4

vector product of two vectors

Answer 4

determinant of the two vectors as a matrix

Question 5

what does the vector product find

Answer 5

a vector perpendicular to the two other vector

Question 6

general vector equation of a line

Answer 6

position vector + λ * direction vector

Question 7

finding the Cartesian equation of a line

Answer 7

set each line equal to x, y or z. then rearrange for λ.

Question 8

finding the intersection of two lines

Answer 8

• equate the two equations
• write down two equations in λ and μ
• solve the equations simultaneously
• substitute back into the original equations to get the positions

Question 9

intersection of lines in 3D

Answer 9

• can be parallel
• can intersect
• can be skew

Question 10

vector equation of a plane

Answer 10

position vector + λ * direction vector + μ* direction vector

Question 11

finding the Cartesian equation of a plane

Answer 11

• vector product the two direction vectors to find the normal
• find the constant by putting in the point is n1x + n2x + n3x = d

Question 12

finding the normal equation of a plane

Answer 12

• find the normal (vector product the two direction vectors)
• find the constant by scalar producting a point and the normal
• write in the form r.n = d (where r stays as r)

Question 13

how can a plane and a line intersect

Answer 13

• intersects once
• line parallel to plane (no points)
• line is within the plane

Question 14

angle between to planes

Answer 14

the angle between the two normal lines

Question 15

distance between a point (b) and a plane (in normal form r.n = p)

Answer 15

D = |(b.n)-p|/|n|

Question 16

point (x,y) and a line (ax+by = c) in 2D

Answer 16

|ax + by - c|/√a^2+b^2

Question 17

cartesian equation of a line in 2D from vector equation

Answer 17

• set the line equal to (x, y)
• rearrange the two equations for λ
• set the equations equal and rearrange into the normal form

Question 18

distance between skew lines

Answer 18

D = |(b-a).n|/|n|

Question 19

When finding an unknown line which intersects planes

Answer 19

let x = λ, find y and z in terms of lambda then put line in cartesian form

Question 20

shortest distance to a point to a line in 3D

Answer 20

|(p-a)xd|/|d|

p is the point, a is the lines position vector, d is the direction vector

Question 21

angle between plane and line

Answer 21

cosθ = n.d/|n||d|
90-θ
draw picture to help understand

Question 22

distance between skew lines

Answer 22

|(b-a).n|/|n|

b and a are the position vectors, n is the cross product of the direction vectors

Question 23

proving the lines are skew

Answer 23

• prove they don’t intersect

- prove they’re not parallel

Question 24

distance between point and plane

Answer 24

|(b.n)-p|/|n|

b is the position vector of the point, n is the normal to the plane, p is RHS of r.n = p

Question 25

distance between point and plane cartesian

Answer 25

|ax+by+cz + d|/√(a^2 +b^2 + c^2)

where a, b, c and d is the cartesian equation, and x,y,z is the point

Question 26

vector product of parallel vectors

Answer 26

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