Difference between two vectors

[happykamper21]

Homework Statement

In most problems involving projections I'm given a vector and the equation of a line either in parametric form or in symmetric form (ie. parametric: <0t+3, -t-4. 3t+2> or symmetric form: x=3, (y+4)/-1, (z-2)/3). However, when asked to use these in a problem I get confused often. So my question is what does <0, -1, 3> and <3, -4, 2> each stand for? When is it appropriate to use one vector over the other?

 

[Mark44]

In most problems involving projections I'm given a vector and the equation of a line either in parametric form or in symmetric form (ie. parametric: <0t+3, -t-4. 3t+2> or symmetric form: x=3, (y+4)/-1, (z-2)/3). However, when asked to use these in a problem I get confused often.

These are different forms of the equation of a line. The parametric form represents the sum of a vector from the origin to the point (3, -4, 2), which is a point on the line, and a vector from that point to an arbitrary point on the line.

Your second form is different from what I've seen. The usual symmetric form would be $\frac {x - 3} 1 = \frac{y + 4}{-1} = \frac{z - 2} 3$. This form also contains information about a vector with the same direction as the line; namely <1, -1, 3> and a point on the line; namely, the point (3, -4, 2).

So my question is what does <0, -1, 3> and <3, -4, 2> each stand for? When is it appropriate to use one vector over the other?

Your examples are vectors. Both can be considered to start at the origin. The first vector you wrote has an endpoint at (0, -1, 3), and the other one ends at the point (3, -4, 2).

Do not confuse lines with vectors -- they are different things. Related, but still different.

 

[happykamper21]

These are different forms of the equation of a line. The parametric form represents the sum of a vector from the origin to the point (3, -4, 2), which is a point on the line, and a vector from that point to an arbitrary point on the line.

Your second form is different from what I've seen. The usual symmetric form would be $\frac {x - 3} 1 = \frac{y + 4}{-1} = \frac{z - 2} 3$. This form also contains information about a vector with the same direction as the line; namely <1, -1, 3> and a point on the line; namely, the point (3, -4, 2).

Your examples are vectors. Both can be considered to start at the origin. The first vector you wrote has an endpoint at (0, -1, 3), and the other one ends at the point (3, -4, 2).

Do not confuse lines with vectors -- they are different things. Related, but still different.

Thank you for such a detailed reply! I was also wondering why my teacher calls <0, -1, 3> a directional vector and what significance it holds when using it in problems such as projections or finding lines parallel to planes?

 

[Mark44]

The two types of vectors you're likely to come across in your class are displacement vectors and position vectors.
Per wikipedia, a displacement vector is "a vector that specifies the change in position of a point relative to a previous position"
A position vector is "a vector representing the position of a point in an affine space in relation to a reference point"

They don't list directional vectors, possibly it's redundant to add "directional." A vector already has a direction, unless it's the zero vector.

For the problems you're working on, I think it's more important that you understand the difference between a vector and a line, which can be defined by a point on the line and a vector having the same direction as the line.

 

[vela]

Thank you for such a detailed reply! I was also wondering why my teacher calls <0, -1, 3> a directional vector and what significance it holds when using it in problems such as projections or finding lines parallel to planes?

The vector <0, -1, 3> specifies the direction of the line, i.e., the line is parallel to that vector. The other vector, <3,-4,2>, is a point on the line. If you change the first vector, the resulting line would still pass through the point <3,-4,2>, but you would be changing the orientation of the line. If you kept the first vector but changed the second vector, the resulting line would be parallel to the original line but translated to pass through the new point.