Vectors

Question 1

How do you find the magnitude of a vector? e.g |a|

Answer 1

(x² + y² + z²)^1/2

Question 2

How do you know if two vectors are parallel?

Answer 2

If the vectors are scalar multiples of one another

Question 3

What is a position vector?

Answer 3

A vector that’s position is described with relation to the origin. e.g OP

Question 4

What is the vector equation of a 3d line?

Answer 4

r = a + λb

Where:
* r is a position vector of a point on the line
* a is a position vector connnecting to the line
* b is a vector that travels along the line (between two points)

Question 5

How do you know if a point lies on a line, given the vector equation of the line and the point?

Answer 5

If once you sub the point in for r your values of λ are consistent the point lies on the line

Question 6

What is the cartesian equation of a vector line?

Answer 6

(x - a₁)/b₁ = (x - a₂)/b₂ = (x - a₃)/b₃

Where:

A = (a₁,a₂,a₃) and is any point on the line

b = (b₁i + b₂j + b₃k) and is any vector parallel to the line

Question 7

How do you convert from a vector equation of a line to a cartesian equation of a vector line?

Answer 7

You subtract the start (the ‘a’ vector) and divide the direction (the ‘b’ vector)

Question 8

What are the three cases for two intersecting lines in 2d?

Answer 8

If the lines are parallel and are not the same line their is no intersection points (no solutions)

If the lines are the same line there are infinitely many intersection points (infinite solutions)

If the lines intersect at a single point there is one intersection point (one solution)

Question 9

How do you multiply vectors?

Answer 9

Do the dot product of the two vectors

Question 10

How do you do the dot product between two vectors? e.g (a₁i + a₂j + a₃) · (b₁i + b₂j + b₃)

Answer 10

a · b = a₁b₁ + a₂b₂ + a₃b₃

Your answer should always be a number

Question 11

What is the formula to find the angle between two vectors?

Answer 11

cos(θ) = (a · b)/(|a||b|)

Where a and b are direction vectors of the two vector lines

If you get an acute angle but it asks for the obtuse angle or vice versa do 180 - first solution

Question 12

How do you know if two vectors are perpendicular? e.g r = a+λb and r = c+λd?

Answer 12

If the dot product of the two vector lines direction vectors equals zero

b · d = 0

Question 13

How do you find the shortest distance from a position vector to a vector line? e.g the shortest distance from OP to the line r = a +λb where OQ is the point on the line closest to OP

Answer 13

• State the point on the line closest to your position vector is OQ
• State that OQ has general position vector e.g ((1+3λ) i + (2-5λ)j +(5+2λ)k)
• State the value of PQ from PQ = OQ - OP
• You know for Q to be the shortes distance from P to the line PQ must be perpendicular to the line so PQ · b = 0
• Solve for λ and sub into equation for OQ to find the point

Question 14

How do you find the shortest distance between two parallel vector lines? e.g r = a+λb and r = c+νd

Answer 14

• State a known point on one of the lines and call it OP
• State a general position vector on the other line e.g ((1+3λ) i + (2-5λ)j +(5+2λ)k) and call it OQ
• You know PQ = OQ - OP
• You know for PQ to be the shortest distance between two lines it must be perpendicular to both lines direction vectors
• Therefore PQ · ‘b or a’ = 0 and solve for λ or ν
• Sub λ or ν back in to your equation for PQ or OP
• |PQ| is now the shortest distance between the lines

Question 15

How do you find the shortest distance between two non parallel vector lines? e.g r = a+λb and r = c+νd

Answer 15

• Let P be a point on line 1 and Q be a point on line 2 such that PQ is the shortest distance between the two lines
• Write OP and OQ as the general position vectors of their respective lines
• You know PQ = OQ - OP
• For PQ to be the shortest distance between the two lines PQ · b = 0 and PQ · ν = 0 solve the two equations you get simultaneously to find the values of ν and λ
• Sub these values into your equation for PQ
• |PQ| is now the shortest distance between the lines

Question 16

What is the vector equation of a plane?

Answer 16

r = a +λb + μc

Where:

• r is a position vector of a point on the plane
• a is a position vector connnecting to the plane
• b is a vector that travels along the plane (between two points)
• c is a vector that travels along the plane (between two points)

Question 17

What is the scalar dot product equation of a plane?

Answer 17

r · n = k

Where:
* r is a position vector of a point on the plane
* n is a normal to the plane
* k = a · n where a is a position vector on the plane

Question 18

How do you know if a point lies on a plane given the vector equation of the plane and the point?

Answer 18

Sub the point in for r into the vector equation of the plane. If the values of λ and μ are consistent then the point lies on the plane

Question 19

How do you know if a point lies on a plane given the scalar dot product equation of the plane and the point?

Answer 19

Do the dot product of the normal to the plane and the point. If the value of your dot product is equal to k then the point does lie on the plane

Question 20

What is the cartesian equation of a vector plane?

Answer 20

ax +by +cz = d

Where:
* n = (ai +bj +ck) and n is the normal to the plane
* r = (xi +yj +zk)
* and d = a · n where a is a position vector on the plane

Question 21

If you require the normal (ai +bj +ck) to a plane but you only have two simultaneous equations e.g 2a + b - c = 0 and a - 3b + 2c = 0 what do you do?

Answer 21

Set a = 1 and attempt to solve the two simultaneous equations for b and c. If you get no solutions a = 0 and set b = 1 and try to solve again, if again you get no solutions b = 0 aswell as ‘a’ and c = 1

Question 22

When converting from vector equation of a line to cartesian equation of a vector line what do you do if one or two of the components for the direction vector of the line is 0?

Answer 22

Instead of (x - a₁)/b₁ = (x - a₂)/b₂ = (x - a₃)/b₃
the terms that have the b component that is 0 become
x - a₁ = (x - a₂)/b₂ = (x - a₃)/b₃

Question 23

What are the three possible cases for intersecting lines and planes?

Answer 23

• If the equations have no solutions then the line is parallel to the plane and doesn’t intersect it
• If the equations have one solution then the line intersects the plane at one point
• If the equations have infinitely many solutions then the line lies along the plane

Question 24

How do you find the point of intersection between two lines?

Answer 24

• Once both lines are in vector equation form equate the equations as they both are r =
• Then solve these simultaneously for the values of μ and λ. Sub the values of μ or λ back into the equation for one of the lines to find they point they intercept at

Question 25

How do you find the point of intersection between a line and a plane given they are both in vector equation form?

Answer 25

• Once both the plane and the line are in vector equation form set both equations equal to eachother through r =
• Solve these simultanously for λ, μ and ν. Sub these values back into one of the equations to get the point they intersect at

Question 26

How do you find the point of intersection between a line and a plane given the plane is given in scalar product form and the line is given in vector equation form?

Answer 26

Sun the general position vector of the line into the equation of the plane for ‘r’ and solve. K = your expression in terms of lamda. Solve for lamda

Question 27

How do you find the point of intersection between a line and a plane given they are both given in cartesian vector form?

Answer 27

• Rewrite the cartesian equation of the line into vector equation form and the general position vector of any point on the line as a single column vector where the components are equal to x, y and z
• Sub these values into the plane equation and solve for the λ, μ or ν
• sub this back into one of the equations to find the point of intersection

Question 28

What can you do if given multiple planes (normally 3) in matrix form?

Answer 28

• Find the determinant of the matrix (if the determinant = 0 they either have no solutions or infinite)
• Solve the matrix equation to find possible points of intersection

Question 29

How do you find the angle between two planes?

Answer 29

cos(θ) = (n₁ · n₂)/(|n₁||n₂|)

Where n₁ and n₁ are normals to the two planes

If you get an acute angle but it asks for the obtuse angle or vice versa do 180 - first solution

Question 30

How do you find the angle between a plane and a line?

Answer 30

sin(θ) = (n · b)/(|n||b|)

Where n is a normal to the plane and b is the direction vector of the line

If you get an acute angle but it asks for the obtuse angle or vice versa do 180 - first solution

Question 31

How do you find the shortest distance (perpendicular) between a point and a plane?

Answer 31

|(n₁a + nb₂ + nc₃ + d) ÷ (n₁² +n₂² +n₃²)^1/2|

Point: (a,b,y)
PLane: n₁x + n₂y + n₃z + d = 0

Note the ‘d’ must be subtracted