Vectors

Question 1

Intersection of two planes

Answer 1

x+2y+z−1=0
2x+3y−2z+2=0
Solve simultaneously to get x and y in terms of z

x = -7 + 7t, y = 4 - 4t

The equation of the line is (-7 + 7t, 4 - 4t, t), or (-7, 4, 0) + t(7, -4, 1)

Question 2

Plane/line intersection

Answer 2

Get x,y,z in terms of lambda
Substitute into the plane equation and solve for lambda
Put lambda back into the line equations

Question 3

Plane containing 3 points

Answer 3

Get 3 simultaneous equations in terms of a, b, and c, all equaling 1
So (4, 3, 1) gives 4a + 3b + c = 1
Solve, and then substitute a, b, and c into ax + by + cz = 1

Question 4

Cross product of (abc) and (xyz)

Answer 4

bz - yc
cx - az
ay - bx

bzc xay

Question 5

Finding the normal vector to a plane containing 3 points

Answer 5

Work out AB and AC
Find the normal vector (abc) to both
Get 2 equations in a, b and c equaling zero
Choose a random value for a, get b and c
Multiply this vector until you get integers
Vector normal to the plane is 1/sqrt(a^2 + b^2 + c^2) (abc)

This is the normal to two lines, both in the plane

Question 6

Area of a triangle with vectors a and b

Answer 6

0.5|a × b|

Question 7

Area of a triangle with vertices a, b, and c being position vectors

Answer 7

0.5|(a×b)+(b×c)+(c×a)|

Question 8

Volume of a parallelopiped or tetrahedron with sides being vectors a, b and c

Answer 8

|a.(b×c)| if parallelopiped

Divide by 6 if tetrahedron

Question 9

What are the direction cosines of (abc)?

Answer 9

cos(x) = a/magnitude
cos(y) = b/magnitude
cos(z) = c/magnitude
X, y and z are the angles that the vector makes with each axis

Question 10

Angle between line r = a + tb and plane r.n = k

Answer 10

sin(x) = |b.n|/|b||n||

Question 11

Angle between planes r.a = b and r.c = d

Answer 11

cos(x) = |a.c|/|a||c||

Question 12

Distance from line (points B and C) to point A

Answer 12

|(BA)x(BC)|/|BC|

Question 13

Shortest distance between two skew lines r = a + tb and r = p + tq

Answer 13

(a - p) . (b x q)/| b x q | |

Question 14

Forms of a plane equation

Answer 14

r = a + Qb + Rc
a is the vector from the origin to the plane
Q and R are vectors in the plane
Normal vector is Q x R

(r-a).n = 0
or r.n = a.n
a is a point in the plane
n is the normal vector to the plane

Question 15

Equation of normal to plane r.m = k passing through a point P

Answer 15

P cross M

Question 16

How do you tell if two lines are in the same plane?

Answer 16

They have a point of intersection

Question 17

Reflection of a line in a plane

Answer 17

Get 2 points on the line
For each point, find the shortest line to the plane (The direction of the line should be the normal to the plane, or the first line’s direction cross a vector in the plane)
Work out the lambda at which this line hits the plane
Double this value to get the 2 points’ reflections in the plane
Join these 2 points to make a new line