Explain why there are many different vector and parametric equations for a plane.

Provide specific examples to justify your answer. (3 marks)

True or False: Two lines must be parallel if they have the same
direction vectors.

A plane can be defined by two direction vector and a point on the plane.

The two direction vectors would "stabilize" the plane and you then move it parallel
to itself to go through the given point.
e.g.
Suppose you have a plane x + y + z = 10
3 points would be (1,0,9), (3,3,1) and (-2,5,7)
so two direction vectors would be <2, 3,-8> and <-5,2,6>
using the point (3,3,1) one possible vector equation would be
r = (3,3,1) + t(2,3,-8) + s(-5,2,6)
the corresponding parametrics are:
x = 3 + 2t - 5s
y = 3 + 3t 2s
z = 1 - 8t + 6s

of course we could have used one of the other two points, giving us two
more versions of equations for the plane
another set of 3 points could be (0,0,10), (-4, 8, 6) and (5,5,0)
we could find totally different direction vectors and using any of the points
we could find several different vector and parametric equations of the same plane

b) T or F:
What does the phrase "same direction vectors" mean to you

sorry, I have a typo

the point (3,3,1) is not on the plane I gave, should have been (3,3,4)
Unfortunately, this messes up both of the direction vector that I used.
Fortunately , my reasoning and argument are still valid, I am sure you can
correct my typo and the results that follow.

Thank you very much

There are many different vector and parametric equations for a plane because there are multiple ways to describe the same mathematical object. Each equation provides a different perspective or approach to representing the plane in terms of vectors or parameters.

One example of a vector equation for a plane is the general form:

r = a + s * u + t * v

where "r" represents a point on the plane, "a" is a known point on the plane, "s" and "t" are scalar values, and "u" and "v" are direction vectors of the plane. This equation allows us to describe any point on the plane using a combination of the known point and the direction vectors.

Another example is the normal vector form of a plane equation:

n · (r - a) = 0

where "n" is the normal vector of the plane, "·" represents the dot product, "r" represents a point on the plane, and "a" is a known point on the plane. This equation states that the dot product of the difference between any point on the plane and a known point on the plane with the normal vector is equal to zero. This allows us to express the plane equation in terms of the orthogonal relationship between the normal vector and the position vectors.

As for parametric equations, one example is the Cartesian coordinate form:

x = a + s * u_x + t * v_x
y = b + s * u_y + t * v_y
z = c + s * u_z + t * v_z

where "x", "y", and "z" represent the coordinates of a point on the plane, "a", "b", and "c" are known constants, "s" and "t" are scalar values, and "u" and "v" are direction vectors of the plane. This parametric equation allows us to express the coordinates of any point on the plane in terms of the known constants, scalar values, and direction vectors.

In summary, there are many different vector and parametric equations for a plane because each equation offers a unique representation or perspective, allowing for different approaches to describe the plane mathematically.

As for the statement "Two lines must be parallel if they have the same direction vectors," it is true. Direction vectors represent the direction of a line. If two lines have the same direction vectors, it means they are pointing in the same direction and hence are parallel to each other. Conversely, if two lines are parallel, their direction vectors will be equal or proportional. Thus, having the same direction vectors is a sufficient condition for lines to be parallel.