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Vector equation when a line passes through a and is parallel to b
r = a + λb
Vector equation when a line passes through c and d
r = c + λ(c - d)
Cartesian equation of line r = a + λb
x - a₁ ÷ b₁ = y - a₂ ÷ b₂ = z - a₃ ÷ b₃
Vector equation of a plane
r = a + λb + µc
The normal vector equation
n = ai + bj + ck
What is the normal?
The direction of the plane that is also perpendicular to it
Cartesian equation of a plane
ax +by + cz = d
Scalar product of two vectors
a.b = |a||b|cosθ
where a and b are position vectors
Scalar product of two vectors that are perpendicular
a.b = 0
where a and b are position vectors
Scalar product of two vectors that are parallel
a.b = |a||b|
where a and b are position vectors
Acute angle between two intersecting straight lines
cosθ = | a.b ÷ |a||b| |
where a and b are direction vectors of the lines
Scalar product form of the equation of the line
r.n = a.n
where r is a position vector
where a is a position vector
where n is the normal
Acute angle between line and plane
sinθ = | b.n ÷ |b||n| |
where b is the direction vector of the line
where n is the normal of the plane
Acute angle between two planes
cosθ = | n₁.n₂ ÷ |n₁||n₂| |
where n is the normal of the two planes
Steps to finding the points of intersections of a line:
Write the line equations in column notation and then equate the equations
Solve the first 2 linear equations
If the first two equations have no solutions, the lines do not intersect. But if they do then check to see if it satisfies the third equation by subbing the values λ and µ into the third equation
The lines intersect if the values of λ and µ satisfy the third equation. If they don’t, then the lines do not intersect
Determining if a line and plane intersect
Work out the dot product of the line and the plane to find λ
If there is one solution, sub λ back into the line equation to get the point of intersection
If there is infinite solutions or no solutions then the line is parallel to the plane
Skew
Two straight lines that are not parallel AND do not intersect
Finding the perpendicular of two parallel lines
Subtract the vector A and B to get the vector AB
Replace λ and µ with t
Find the dot product of the vector AB with the direction vector to get t
Sub t into AB vector to get the shortest distance
Finding the perpendicular of any two lines
Subtract the vector A and B to get the vector AB
Find the dot product of the vector AB with the direction vectors A and B to get simultaneous equations
Solve to get λ and µ
Sub λ and µ back into the vector AB to get the shortest distance
Shortest distance between the origin and the plane
r.unit vector = k
where k is the length of shortest distance from the origin to the plane
unit vector = n ÷ |n|
Shortest distance from a point to a plane equation
|n₁α + n₂β + n₃γ - d| ÷ √(n₁² + n₂² + n₃²)
Steps to solving reflection of lines in planes
Find the dot product of the line vector equation AP and the plane vector equation to get λ
Sub λ into the vector equation AP to get the point A
Find P when λ = 0
Form the vector equation of PQ using the point P and the direction vector of the plane
Find the dot product of the line vector equation PQ and the plane vector equation to get µ
Sub µ into the vector equation PQ to get the point Q
Then subtract the point Q from A to get the direction vector for the reflected line
The equation of the reflected line is either point A or Q and the direction vector found
Shortest distance between point and line
Subtract the vector equation of the line A with the point B to get the vector AB
Find the dot product of AB and the direction vector of the line to find λ
Sub λ into AB to get the shortest distance
Collinear
All points lie on the same line
Coplanar
All points lie on the same plane
How to show that points A, B and C are collinear
Find the vector AB and AC
If they are multiples of each other then they are parallel and A is common in both so they are collinear
How to find d in the equation of a plane
sub in the point A into the equation to find d
How to verify that a plane passes through a point
Sub in the point A into the plane’s equation
If it equals d, then the planes passes through that point
How to verify points are coplanar?
Subtract the point A from B and C to get two direction vectors
Use this to create an equation for the plane
Then equate the plane’s equation to point D to solve for λ and µ
Point D lies on the plane if the values of λ and µ satisfy the third equation
