Vectors

Further Maths

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:

1. Write the line equations in column notation and then equate the equations

2. Solve the first 2 linear equations

3. 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

4. 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

1. Find the dot product of the line vector equation AP and the plane vector equation to get λ

2. Sub λ into the vector equation AP to get the point A

3. Find P when λ = 0

4. Form the vector equation of PQ using the point P and the direction vector of the plane

5. Find the dot product of the line vector equation PQ and the plane vector equation to get µ

6. Sub µ into the vector equation PQ to get the point Q

7. Then subtract the point Q from A to get the direction vector for the reflected line

8. The equation of the reflected line is either point A or Q and the direction vector found

Shortest distance between point and line

1. Subtract the vector equation of the line A with the point B to get the vector AB

2. Find the dot product of AB and the direction vector of the line to find λ

3. 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