Vectors Analysis Flashcards
How to prove two vectors are perpendicular to each other
If the dot product of the vectors equals zero
a . b = 0
How to prove two vectors are parallel to each other
If the cross product of the vectors equals zero
a x b = 0
Dot Product
a . b =
|a||b| cos θ
(ax bx + ay by + az bz)
- gives the direction of one vector relative to another
Cross Product
a x b =
|a||b| sin θ ň
|a x b| = |a||b| sin θ
i (aybz- azby) - j (axbz - azbx) + k (axby - aybx)
- gives angle
- can be used to calculate the area by multiplying by 1/2
Scalar Triple Product
(a x b) . c =
(aybz - azby) cx + (azbx - axbz) cy + (axby - aybx) cz
answer is just a number
used to calculate the volume
ax ay az |
|bx by bz |
What is cross product also known as
Vector Product
How do you prove vectors are coplanar to each other
If the scalar product of the vectors equals zero
(a x b) . c = 0
What is dot product also known as
Scalar Product
Express a vector in the Cartesian form
vector = i + j + k
Express a vector in the row vector notation
vector = (x, y, z)
Calculate the unit vector û
= u / |u|
= i + j + k / sqrt(i^2 + j^2 + k^2)
Unit vectors specify the direction of a vector
gradient of a scalar field
∇φ = dφ / dx î + dφ / dy ĵ + dφ / dz k̂
scalar vector (φ), (x, y, z)
differentiate the whole vector three times in terms of x y z then multiple by i j k
directional derivative (divergence) of a vector field
= gradient x unit vector = ∇φ * û
should produce a scalar answer (only numbers)
When does the minimum directional derivative occur
when the maximum rate of change cosφ is -1, when φ = 180
where u points in the opposite direction to ∇φ
maximum rate of increase and decrease
Increase:
= |∇φ| = sqrt(x^2 + y^2 + z^2) = sqrt(gradient)
Decrease:
= - |∇φ|= - sqrt(x^2 + y^2 + z^2)
find unit vector perpendicular to two vectors
find ñ
maximum rate of change
= dφ / ds = |∇φ|cosθ
determine the expression for divA, ∇.A
= da / dx + db / dy + dc / dz
differentiate separate in terms of x y z
curl ∇ x B
shows a vector field’s rate of rotation (direction of the axis of rotation)
= cross product of the gradient
A = a î + b ĵ + c k̂
curl A =
|| î ĵ k̂ |
| d/dx d/dy d/dz |
| a b c |
(laplace expansion)
what is a vector field with zero curl
irrotational
