Intro lecture

Question 1

what is the dot product of two vectors a (a1, a2, a3) and b (b1, b2, b3)?

Answer 1

a.b =a1b1 + a2b2 + a3b3

Question 2

what is the dot product of two vectors a and b, given their magnitudes and the angle between them

Answer 2

a.b = |a||b|cos (theta)

Question 3

what two properties does the dot product have?

Answer 3

linearity: (alphaa + betab).c = alphaa.c + betab.c
symmetry: a.b = b.a

Question 4

how do you calculate the magnitude of a vector? (otherwise known as norm or length)

Answer 4

|a| = \sqrt(a.a) = \sqrt(a1^2 + a2^2 + a3^2)

Question 5

how can you tell if two vectors are orthogonal?

Answer 5

their dot product is zero

Question 6

how can you give the equation of a plane, given a vector that is normal to it (a) and any point (r) on the plane (P)?

Answer 6

r.a = constant
this is because if r0 is a fixed point on the plane, then r-r0 must definitely be perpendicular to a, so (r-r0).a=0, so r.a=r0.a=constant

Question 7

how can you easily check if a plane passes through the origin

Answer 7

it’s constant will be equal to zero, when you determine the coefficients of (x,y,z) by the vector a, and the values of (x,y,z) by a point r0

Question 8

how can you write the equation of a plane in a way that looks cartesian?

Answer 8

a.r = a1x + a2y + a3z = constant
(with r = (x,y,z) )

Question 9

what does the cross product give you?

Answer 9

a vector denoted a x b that is orthogonal to both a and b and thus orthogonal to the plane spanned by a and b

Question 10

how do you calculate the magnitude of the cross product?

Answer 10

|a x b| = |a||b|sin(theta_{ab})

Question 11

to work out the direction of the cross product, which fingers equate to which vectors in the right hand rule?

Answer 11

a is the first finger
b is the middle finger
a x b is the thumb

Question 12

what is the magnitude of the cross product equal to

Answer 12

|a x b| is the area of the parallelogram that is spanned by the two vectors a and b

Question 13

what are the two properties of the cross product?

Answer 13

linearity: a x (alphab + betac) = alphaa x b + betaa x c
anti-symmetry: b x a = - a x b

Question 14

what is the cross product of a vector with itself?

Answer 14

the zero vector, i.e. a x a = 0

Question 15

what is the cross product of a vector with any parallel vector and why?

Answer 15

0, as sin(theta_{ab})=0 with theta_{ab} either zero or pi

Question 16

how do you find the area of a triangle given it’s vertices?

Answer 16

using one point as a basis, determine the vectors toward each other point (i.e. (B-A) and (C-A))
the magnitude of the x product gives the parallelogram area of these vectors
divide this area by 2 to get the triangle area

Question 17

given three vectors, a,b and c in R^3, what is the scalar triple product?

Answer 17

(a x b) . c

Question 18

properties of the scalar triple product

Answer 18

cyclically symmetric:
i.e. a.(bxc) = b.(cxa) = c.(axb)
pairwise anti-symmetric:
[a,b,c] =-[a,c,b]

Question 19

how do you find the volume of a parallelepiped given 3 vectors?

Answer 19

compute the determinant of the 3 vectors

Question 20

what is the vector triple product?

Answer 20

a x (b x c)

Question 21

what is the vector triple product equal to?

Answer 21

a x (b x c) = (a.c)b - (a.b)c

Question 22

what is a scalar field?

Answer 22

an assignment of a number (scalar) to every point in space
e.g. T(x) is the temperature at every point in a room

Question 23

what is a vector field

Answer 23

an assignment of a vector (usually in R^3) to a every point in space

Question 24

how might we write the vector field A(r) which is in R^3?

Answer 24

A(r)=A_x(r)i + A_y(r)j +A_z(r)k
where the components A_x, A_y and A_z are scalar functions of the position vector r = xi + yj + zk

note that the symbol r is commonly used for the position vector in R^2 or R^3