Vectors Flashcards
How to find a unit vector
Unit vector = vector / absolute value of a vector
Definition of scalar product
a . b = |a||b|cos φ
What is the geometrical interpretation of a . b
- Projecting the line a onto the line b
How do you calculate the scalar product of 2 vectors
- ## Multiply each term in a by its corresponding term in b and then sum the products. It will give a scalar
What are the 6 basic rules for the scalar product
What is the vector product equation
a x b = |a||b|sinψ η(unit)
What is special about the vector η resulting from the vector product
- It is perpendicular from both a & b
What is the geometrical interpretation of the vector product
- ## The vector product is the area of the parrallogram of a b
What are the basic rules for Vector product
- Anti-Commutative rule
- Non associative multiplication
- Disributive law
What is the matrix method for calculating vector product
What is the alternative method for calculating vector product
What is the triple scalar product
a . (b x c)
How to calculate the triple scalar product.
- Do cross product on the brackets then do dot product on the results
Do the brackets matter for the triple scalar product
No
What is the triple vector product
a x (b x c)
What is the bac cab rule
- The following identity is true for triple vector product
What is the vector equation of a line
Alternative for the vector equation of the line if we have vector c in direction of the line
Cartesian eqaution of a line
How to get from vector equation to cartesian equation
Name the values for λ that have key points of intrest
What to do if the cartresian form of a line eqaution has 0 as a demoninatior
- Solve in terms of x and y in one eqation and z in another
What are the three possible cases for intersection of two lines
- They do not intersect and the system of equations has no solutions
- They intersect at a point whos position vector (x) is given by: {x = a+λc = b+μd} where a+λc and b+μd are the two eqations of lines
- The system has an infinite number of solutions as the lines are the same
How to position of two intersecting vectors
- Set the two eqations equal to each other.
- ## Solve for μ or λ and then sub one of these values into its repective equation
How to find the angle of intersection between two lines
- Use the rearrangement of the scalar product
- Remeber, the angle between the lines is equivilent to finding the angle between the two directional vectors
What is the vector equation of a plane
What is the equation for the shortest distance between a line and a plane
- Where d = p in the plane equation
What is the shortest distance from the plane to an origin
Where a is a point on the plane and n is the unit vector of the normal
How do you find the intersection between a line and a plane
- Plug in the values for the line (e.g 2-λ) into the point on a plane part of the plane.
- Solve for λ
- Using this value of λ and the origional equation of the line, get the point which intersects with the plane
What will the intersection of two planes give us
- A line
How to find the intersection of two planes
- To get the directional vector of the line (b) we do the cross product of the normals of each plane. (As we know the line is perpendicular to both normals)
- To get the position vector a: we know that this point is on both lines so we solve for a position that fufills both planar equations. This can include picking a random number.
How to find the angle between two planes
- ## Use the scalar product rearragned on the normals of both planes
How to find the angle between a line and a plane
- We can easily find the angle between a line and the normal using the rearranged scalar product
- ## We can then subtract this from 90 degrees to find the angle between the line and the plane
How to find the vector equation of a plane from 3 points
- We have points A, B, C in a plane
- Do the cross product of the lines AB and BC.
- This will yield the normal to the plane
- Now using the equation for a plane we do the dot product of this normal and a point on the plane to get p
