Vectors

Question 1

Vector cross product

Answer 1

^
a x b = |a| |b| sinθ n
θ is the angle between vectors a,b
|a| |b| sinθ is the magnitude of the vector
Finds a vector perpendicular to a and b

Question 2

Does the order matter cross product

Answer 2

Yes, a x b = -(b x a)

Question 3

Right-hand rule

Answer 3

Thumb, index, middle fingers at right angles
Thumb = a x b
Index = a
Middle = b

Question 4

Conditions for a x b = 0

Answer 4

a,b are parallel or one is 0

Question 5

i x j

Answer 5

k

Question 6

k x j

Answer 6

-i

Question 7

i x k

Answer 7

-j

Question 8

Cross product method for vectors a(a1, a2, a3) and b(b1, b2, b3)

Answer 8

i j k |
| a1 a2 a3 |
| b1 b2 b3 |

Question 9

Area of a triangle from 2 directions

Answer 9

1/2 |a x b|

Can find a,b if you have 3 points

Question 10

Area of a parallelogram from the 2 directions

Answer 10

|a x b|
a, b are not parallel sides
Can find a,b with vectors between vertices

Question 11

a • (b x c)

a(a1, a2, a3), b(b1, b2, b3), c(c1, c2, c3)

Answer 11

a1 a2 a3 |
| b1 b2 b3 |
| c1 c2 c3 |

Question 12

Triple scalar product facts

Answer 12

Reordering a, b, c has no effect as long as a,b,c cycle is maintained, else result is negated
a. (a x b) = 0

Question 13

Volume of a parallelepiped

Answer 13

|a • (b x c)|
h = |a| cosθ (θ is angle between vertical and edge with vertical component)
area of base = |b x c|
V = |b x c| |a| cosθ

Question 14

Parallelepiped

Answer 14

A prism with parallelograms for all 6 faces

Question 15

Equation of a line with cross product

Answer 15

r x b = a x b (r is a general point on the line, a is a known point on the line, b is direction)

Question 16

Line equation derivation

Answer 16

r - a = b (from known point)
r = a + λb (from general point)
b is parallel to λb so (r- a) x b = 0
expand tp r x b = a x b

Question 17

Cosines

Answer 17

cos^2 α + cos^2 β + cos^2 γ = 1

α, β, γ are the angles between the vector a and the i, j, k axes respectively

Question 18

Equation of a plane cross product method

Answer 18

Identify two vectors parallel to the plane
n is the cross product of these
r . n = a . n

Question 19

Intersection of planes

Answer 19

Use cross product to find a vector perpendicular to both planes (direction, d)
Let x = 0 and solve the plane equations simultaneously for a point (a)
Use r = a + λd

Question 20

Shortest distance between skew lines r = a + λb and r = c + μd

Answer 20

(a - c) • (b x d) |
| ——————– |
| | b x d | |

Question 21

Shortest distance between parallel linesr = a + λb and r = c + μb

Answer 21

b x (a - c) |
| ————–|
| |b| |

Question 22

Volume of a tetrahedron

Answer 22

1/6 |a • (b x c)|