Vectors

Question 1

Vector/parametric eqn of a plane

Answer 1

π: r = a + λb + μc

any pt on plane = position vector + 2 non-zero, non-// directions

Question 2

Eqn of plane in scalar prod form

Answer 2

π: r . n = d, where d = a . n, is a scalar

position vector . normal vector

Question 3

Cartesian eqn of a plane

Answer 3

replace r in r . n = d with (x y z) - column vector

π: n1x + n2y + n3z = d

Question 4

If you get y + 3z = 5 what do u write for the missing x for cartesian eqn

Answer 4

y + 3z = 5, x ∈ R

Question 5

Suggest a few pts that are on the plane r . (2 = 12
-1
3)

Answer 5

2x - y + 3z = 12
eg
Let x+y+0, Z = 4

Question 6

Determining r/s b/w 2 lines

Answer 6

// lines - show b = k d - direction vectors must be //

Intersecting lines - form 3 eqn by equating the x, y & z components of the above vector eqn and solve for λ & μ - must have a unique soln

Skewed lines - not // and do not intersect

Question 7

R/s b/w line and a plane

Answer 7

For l: r = a + λb & π: r . n = d
b . n a . n
l // and doesn’t lie on π: = 0 not d
l // and lies on π: = 0. = d
l not // & intersects π at 1 pt: not 0. -

Question 8

R/s b/w 2 planes

Answer 8

planes are // to each other: normals of planes are //
planes are not // to each other: solve for line of intersection - write out cartesian eqn for both plane and solve using GC

Question 9

Find pt of reflection along a line/plane

Answer 9

ON = (OP + OP’) / 2, where N is the midpt of both pts and OP and OP’ are position vectors

Question 10

Perpendicular dist from pt to line (2)

Answer 10

Foot of perpendicular & projection method

Question 11

Perpendicular dist b/w 2 parallel lines (2)

Answer 11

same as from pt to line - Foot of perpendicular & projection method

Question 12

perpendicular dist b/w line and plane (2)

Answer 12

Foot of perpendicular & projection method

Question 13

perpendicular dist b/w 2 parallel planes (2)

Answer 13

Formula & projection method

Question 14

Angle b/w lines

Answer 14

a.b = cosΘ |a| |b|

Question 15

Angle b/w line and plane

Answer 15

cos (90-Θ) = a.n / |a| |n| OR sinΘ…

Question 16

Angle b/w plane and plane

Answer 16

cos (Θ) = n1 . n2 / |n1| |n2|

Question 17

Ratio theorem

Answer 17

-

also doesn’t have to be from Origin just 1 common pt

Question 18

Find foot of the perpendicular from a pt to a plane

Answer 18

Step 1: Form an eqn of the line passing thro the given pt A and the foot F using the normal vectors as the direction vector, i.e. r = a + λb

Step 2: Find foot F as a pt of intersection of the line and the plane by substituting OF = a + λn into the eqn of the plane to solve for λ

Question 19

Find foot of the perpendicular from a pt to a line

Answer 19

Step 1: F lies on l so u can write OF = a + λb for osme values of λ

Step 2: CF is perpendicular to l, i.e. CF.b = 0 and solve for λ