Vectors

Question 1

How to find a unit vector

Answer 1

Unit vector = vector / absolute value of a vector

Question 2

Definition of scalar product

Answer 2

a . b = |a||b|cos φ

Question 3

What is the geometrical interpretation of a . b

Answer 3

• Projecting the line a onto the line b

Question 4

How do you calculate the scalar product of 2 vectors

Answer 4

• ## Multiply each term in a by its corresponding term in b and then sum the products. It will give a scalar

Question 5

What are the 6 basic rules for the scalar product

Answer 5

Question 6

What is the vector product equation

Answer 6

a x b = |a||b|sinψ η(unit)

Question 7

What is special about the vector η resulting from the vector product

Answer 7

• It is perpendicular from both a & b

Question 8

What is the geometrical interpretation of the vector product

Answer 8

• ## The vector product is the area of the parrallogram of a b

Question 9

What are the basic rules for Vector product

Answer 9

• Anti-Commutative rule
• Non associative multiplication
• Disributive law

Question 10

What is the matrix method for calculating vector product

Answer 10

Question 11

What is the alternative method for calculating vector product

Answer 11

Question 12

What is the triple scalar product

Answer 12

a . (b x c)

Question 13

How to calculate the triple scalar product.

Answer 13

• Do cross product on the brackets then do dot product on the results

Question 14

Do the brackets matter for the triple scalar product

Answer 14

No

Question 15

What is the triple vector product

Answer 15

a x (b x c)

Question 16

What is the bac cab rule

Answer 16

• The following identity is true for triple vector product

b(a.c) - c(a.b)

Question 17

What is the vector equation of a line

Answer 17

Question 18

Alternative for the vector equation of the line if we have vector c in direction of the line

Answer 18

Question 19

Cartesian eqaution of a line

Answer 19

Question 20

How to get from vector equation to cartesian equation

Answer 20

Question 21

Name the values for λ that have key points of intrest

Answer 21

Question 22

What to do if the cartresian form of a line eqaution has 0 as a demoninatior

Answer 22

• Solve in terms of x and y in one eqation and z in another

Question 23

What are the three possible cases for intersection of two lines

Answer 23

• 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

Question 24

How to position of two intersecting vectors

Answer 24

• Set the two eqations equal to each other.
• ## Solve for μ or λ and then sub one of these values into its repective equation

Question 25

How to find the angle of intersection between two lines

Answer 25

- Use the rearrangement of the scalar product - Remeber, the angle between the lines is equivilent to finding the angle between the two directional vectors

Question 26

What is the vector equation of a plane

Answer 26

Question 27

What is the equation for the shortest distance between a line and a plane

Answer 27

- Where d = p in the plane equation

Question 28

What is the shortest distance from the plane to an origin

Answer 28

Where a is a point on the plane and n is the unit vector of the normal

Question 29

How do you find the intersection between a line and a plane

Answer 29

- 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

Question 30

What will the intersection of two planes give us

Answer 30

- A line

Question 31

How to find the intersection of two planes

Answer 31

- 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.

Question 32

How to find the angle between two planes

Answer 32

- Use the scalar product rearragned on the normals of both planes -

Question 33

How to find the angle between a line and a plane

Answer 33

- 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 -

Question 34

How to find the vector equation of a plane from 3 points

Answer 34

- 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