vectors

Question 1

vector equation

Answer 1

r = a + λb

r is a general point
a is a position vector
λ is a scalar
b is a direction vector

Question 2

vector equation that passes through C and D

Answer 2

r = c + λ(d - c)

Question 3

vector equation Cartesian form

Answer 3

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

Question 4

vector equation of a plane

Answer 4

r = a + λb + μc

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

Question 5

cartesian equation of a plane

Answer 5

ax + by + cz + d = 0

(a, b, c) is the normal to the plane
d = -k

Question 6

scalar product form of the equation

Answer 6

r•n = k

where k = a•n for any point on the plane with position vector a

Question 7

scalar/dot product of two vectors

Answer 7

a•b = |a||b|cosθ

cosθ = (a•b) ÷ |a||b| →acute angle

if a and b are non zero vectors then a•b = |a||b|, they are parallel

non zero vectors a and b are perpendicular if a•b = 0

Question 8

acute angle between a line and a plane

Answer 8

sinθ = |(b•n)÷(|b||n|)|

Question 9

acute angle between two planes

Answer 9

cosθ = |(n1•n2)÷(|n1||n2|)|

Question 10

what does it mean if two straight lines don’t intersect

Answer 10

either they are parallel or skew

Question 11

how to determine if two lines meet

Answer 11

1. write equations in column notation and make them equal to each other
2. write three linear equations
3. solve the first two simultaneously
4. if it doesn’t work, no solutions. if it does, see if it works for the third. if it works, they intersect. substitute values to find the point of intersection

Question 12

how to find the length of the perpendicular from the origin to plane Π

Answer 12

r•ñ = k

ñ is a unit vector perpendicular to Π