Further Pure Flashcards
What are the vector equations for the area of a parallelogram or triangle?
Ap=1/2|axb|, At=1/2|axb|
What are the vector equations for the volume of a parallelepiped and tetrahedron?
Vp = a·(bxc), Vt = 1/6(a·(bxc))
What are the generic cartesian and parametric equations for a parabola?
y²=4ax
x=at², y=2at
What is the differential of y²=4ax?
dy/dx= 2a/y
What are the generic cartesian and parametric equations for a rectangular hyperbola?
xy=c²
x=ct, y=c/t
What is the t-formula for sinθ?
sinθ = (2t) / (1+t²)
What’s the t-formula for cosθ?
cosθ = (1-t²) / (1+t²)
What’s the t-formula for tanθ?
tanθ = (2t) / (1-t²)
What is the cross product equation for a straight line?
(r - a) x b = 0
r x b = a x b
How do you work out the cross product manually?
Determinant of:
i j k
a1 a2 a3
b1 b2 b3
REMEMBER to subtract j
What are direction cosines?
The cosine of the angle the direction vector of a line makes with the axis, typically notates as l, m and n, or cos(alpha), cos(beta), cos(gamma).
How do you calculate direction cosines?
a = xi + yj + zk
l = x/|a|
m = y/|a|
n = z/|a|
l² + m² + n² = 1
What is Euler’s method for finding approximate solutions to first order differential equations?
x1 = x0 + h
dy/dx ≈ (y1 - y0)/h
What is the midpoint method for finding approximate solutions to first order differential equations?
(dy/dx)0 ≈ (y1 - y-1) / 2h
What is the method for approximating second order differential equations?
(d²y/dx²) refers to the change in gradient over change in x, so find the difference between the gradient between y-1 and y0 and the gradient between y0 and y1 and divide by h.
This leads to (y1 - 2y0 + y-1)/h²
What is Simpson’s rule?
area = (1/3)(h)(endpoints +4(odd values) + 2(even values))
When determining if a line intersects a plane, what should you remember?
After determining if the line is perpendicular to the normal using the dot product, substitute a value from the line into the plane equation to see if the line lies within the plane or not, to obtain a contradiction.
How do you find the shortest distance between two skew lines?
Find a common normal of the two lines, by doing the cross product of both direction vectors. Find out the constant in the cartesian form of the equation by substituting in a point on the line (… + d = 0), and then use the formula of shortest distance to plane from point as given in the formula book.
How do you determine the coordinates of two points on a line equidistant from another point?
With points B and C on the line and A not on the line, start with |AB| = [distance] to form a quadratic in terms of lambda.
What is the cross product equivalent to?
a x b = |a||b|sinθ ñ
What is the t formula for dθ?
2 / (1+t²)
How do you work out the direction of the vector outputted by the cross product?
Right hand rule
a x b = c
Fingers in order: a, b, c
How can you factorise expressions in two variables, e.g. (p³ - q³)?
- You can use the factor theorem by guessing factors, e.g. p-q=0, so p=q, so p³-q³=p³-p³=0
- You can factorise by dividing by the factor with algebraic division, e.g. (p-q)|p³ + 0p²q + ….
How can you work out the distance to the focus from a point on a conic section?
e(Distance from directrix)
