Y1, C9 - Vectors Flashcards
What is the vector equation r of a straight line
r = a +λb
How do you find the unit vector of a vector (5i, 3j, 2k)
Divide the vector by its magnitude e.g. 1/root(38) * (5i, 3, 2k)
What is the cartesian form of a vector
(x-a1)/b1 = (y-a2)/b2 = (z-a3)/b3 = λ
Convert x-3 / 5 = y+2 / 3 = 4 - z / 1 to vector form
Multiply the z-4 / 1 by -1
(3i, -2j, 4k) + λ(5, 3, -1)
How to convert between vector and Cartesian form of a line
r = a + λb
r = (a1+λb1, a2+λb2, a3+λb3)
(x, y, z) = (a1+λb1, a2+λb2, a3+λb3)
x = a1 + λb1
y = a2 + λb2
z = a3 + λb3
Therefore (x-a1)/b1 = (y-a2)/b2 = (z-a3)/b3 = λ
Convert r = (2, 5, 0) + λ(1, 3, -2) to Cartesian form
(x-2)/1 = (y-5)/3 = (z)/-2 = λ
Convert (x-2)/3 = (y+5)/1 = z/4 to vector from
r = (2, -5, 0) + λ(3, 1, 4)
What is the parametric vector form for a plane
Π = a + λb + μc
What is the Cartesian equation for a plane
Π: n1x + n2y + n3z = c
The plane P is perpendicular to the normal n = 3i - 2j + k and passes through the point with position vector 8i + 4j - 7k, find the cartesian equation of the plane P
3x - 2y + z = c
(8, 4, -7) = x, y, z
sub in
c = 9
Therefore:
3x - 2y + z = 9
Formula for the scalar dot product a.b
a.b = lal * lbl * cosx
What is the dot product of vectors at a right angle (perpendicular)
0
What is the dot product of two vectors with an angle of 0 degrees
lal * lbl
How to find the dot product of two vectors multiplied together
Find the separate dot product of the i, j, and k components and add them
What is the formula for the angle between two vectors
cosx = (a.b) / lallbl
How do you always get an acute angle from the vector angle formula
cosx = modulus of ((a.b) / lallbl)
If I know the points A, B, and C, what vectors do I need to know to find the angle ABC
BA and BC (vectors must come from the angle they are trying to find)
If two vectors are perpendicular what is their dot product
a.b = 0
How would you find a vector which is perpendicular to two points
Multiply both points by (x, y, z) to create simultaneous equations
Set z to 1
Solve for x and y and then multiply out any fractions
If a is a position vector of a point on a plane and so is r, what is (r - a) parallel and perpendicular to
(r - a) is a vector parallel to the plane
(r - a) is perpendicular to n (normal)
What is (r - a) . n
0
What is r.n equal to
r.n = a.n
r.n = p
r.n = p
(x, y, z) . (n1, n2, n3) = p
Therefore what is the Cartesian form of a plane
n1x + n2y + n3z = p
What is the scalar product form for a plane
r.(x, y, z) = p
What are the 3 equations for a plane
r.n = a.n
r.n = p
n1x + n2y + n3z = p
Where n is the normal to the plane, p is a constant, r is the position vector of some point on the plane
n and a are fixed
A point with position vector 2i + 3j - 5k lies on the plane and the vector 3i + j - k is perpendicular, find (a) the Scalar product form, (b) the Cartesian form
a) r.(3, 1, -1) = 14
b) 3x + y - z = 14
How do you find the angle between two lines
Find the angle between their direction vectors
What is the formula for finding the angle between a line and a plane
sinx = modulus((n.b) / lnllbl)
What is the formula for finding the angle between two planes
cosx = modulus((n1.n2) / ln1lln2l)
How can you find if 4 points are coplanar
Find an equation for a plane containing 3 of the 4 points and then investigate if the 4th point is on the plane
What does it mean if two straight lines are skew
They do not intersect and are not parallel
How top check if two lines are parallel
Check if their direction vectors are multiples of each other
Find the intersection of the line l and the plane pi where:
l: r = -i + j -5k + λ(i + j + 2k)
pi: r . (i + 2j + 3k) = 4
(-1 + λ, 1 + λ, -5 + 2λ) . (1, 2, 3) = 4
Solve for λ = 2
Plug in to find point r = (1, 3, -1)
When finding the intersection between a line and a plane you usually find λ = a (where a is a constant), what does it mean if your λ’s cancel and you get:
a) a = a
b) b = a
a) All points of the line lie on the plane, the line is on the plane
b) No points lie on the plane, therefore the line and plane are parallel, LHS not equal to RHS
How can you find the intersection of two planes line
Find 2 common points on both planes, find the vector through them (direction vector) and then form the line equation
When 2 planes intersect, what is the normal to the normals of both planes equal to
The direction of the line
How would you find the vector equation of the line of intersection of the planes r.(2i - 3j + 4k) = 8, and r.(4i + j - 7k) = 0
Solve simultaneously and find two points on the plane then find the vector between them
4x + 7y - 7z = 0
2x - 3y + 4z = 8
Let z = 1, 2 to find 2 points
What is the dot product of the shortest distance between two planes / points / lines
0 (they are perpendicular)
Steps for finding the shortest distance between two lines
1) Find general point on l1
2) Find general point on l2
3) Find vector between them
4) Ensure that this vector is perpendicular to l1 (and l2), by finding the dot product between the direction of the lines and the direction vector between both lines
When finding the shortest distance between two parallel lines, what do we do when we find multiples of mu - lambda
We set mu - lambda to a new variable t
What are the steps to find the shortest distance between a point and a line
1) Find general point on l1
2) Find vector between point and general point (AB = b - a)
3) Ensure the vector is perp to l1 (dot product = 0)
How do you find the shortest distance between a point and a plane
Use the formula in booklet
How do you find the shortest distance between two parallel planes
Find any point on one of the planes
Use the formula for shortest distance between a point and a plane
How would you find any point on the plane 2x - 6y + 3z = 9
Sub in any values which make the equation true e.g. x = 0, y = 0, z = 3
How do you convert between parametric and cartesian form for a plane
Find a vector which is perpendicular to both direction vectors
Let n = (x, y, z) and solve simultaneously to find the normal (n)
r.n = a.n
r.n = (point on the plane) . n
Rearrange to find r
Hence cartesian form found
Convert r = (3i + 4j + k) + λ(4i + k) + μ(8i + 3j + 3k) from parametric form into cartesian form
Let n = (x, y, z)
(8, 3, 3) . (x, y, z) = 0 and
(4, 0, 1) . (x, y, z) = 0
8x + 3y + 3z = 0 and
4x + z = 0
Let z = -4
x = 1, y = 4/3
n = (1, 4/3, -4) = (3, 4, -12)
r.n = a.n so
r.(3, 4, -12) = (3, 4, 1) . (3, 4, -12) = 9 + 16 - 12 = 13
Hence plane is 3x + 4y - 12z = 13
ans = 3x + 4y - 12z - 13 = 0
The plane pi has equation r.(i + 2j + 2k = 5, the point P has coordinates (1, 3, -2), what are my n1, n2, n3, alpha, beta, gamma, -d values for the shortest distance formula?
1 = n1
2 = n2
2 = n3
-5 = d
1 = alpha
3 = beta
-2 = gamma
he plane pi has equation r.(i + 2j + 2k = 5, the point P has coordinates (1, 3, -2), find a) the shortest distance and b) the point of reflection P in pi
a) Use formula to find ans = 2/3
b) Turn the reflection into the intersection of a line and a plane
Find the intersection of the line and the plane:
(1+λ, 3+2λ, -2+2λ) . (1, 2, 2) = 5
λ = 2/9
Double λ and sub in to find reflected point
ans = (13/9, 35/9, -10/9)
How do you reflect a line in a plane
Find the intersection
Reflect a point l1 to l2
Find the equation between 2 points
