Vectors and the Geometry of Space

In summary: Here are some general steps:1) Write the equations of line 1 and line 2 in vector form as R1(t) and R2(s).2) Set R3(u) equal to the given point A, which has coordinates (xo, yo, zo).3) Equate the corresponding components of R3(u) and A to get three equations in three unknowns (t, s, and u).4) Solve for t, s, and u using the three equations. This will give you the specific values of t and s that make it possible for R3 to go through A.5) Substitute those values into the equation for R3 to get the specific equation for the line that goes through A
  • #1
tmlfan_17
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Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.
 
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  • #2
tmlfan_17 said:
Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.
You failed to include equations 1 and 2.
 
  • #3
Equation 1 and 2 are not supposed to be given.
 
  • #4
tmlfan_17 said:
Line 1 and line 2 are given by equation 1 and 2. Point A has coordinates (xo, yo, zo). Find the equation of line 3 which goes through A and crosses L1 and L2.

Suppose line 1 is R1(t) = P + tD1
Line 2 is R2(s) = Q + sD2

where the P and Q are given points and the D's are given direction vectors.

Then a line from one to the other would be
R3(u) = R1(t) + u(R2(s)-R1(t))

Set that equal to your given point. You have three given coordinates of the point and three parameters s,t, and u to work with.
 
  • #5
I don't understand why you put R1(t) as the r0 vector in the R3(u) vector equation instead of just inputting the given point A there.
 
  • #6
tmlfan_17 said:
I don't understand why you put R1(t) as the r0 vector in the R3(u) vector equation instead of just inputting the given point A there.

The line R3 goes from the line with parameter t to the other with parameter s with t and s unknown. You have to choose s and t so that it's possible for R3 to go through A for some value of u. For most choices of s and t the line won't pass through A for any value of u.

[Edit -- Added] The general formula you get is long and messy but doing it for specific lines and points isn't too difficult.
 
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1. What is a vector in the context of geometry?

A vector in the context of geometry is a mathematical quantity that has both magnitude and direction. It is typically represented by an arrow in a coordinate system and can be used to describe the position, motion, or force of an object in space.

2. How are vectors represented and calculated in three-dimensional space?

In three-dimensional space, vectors are typically represented by three coordinates (x, y, z) and can be calculated using vector operations such as addition, subtraction, and scalar multiplication. They can also be represented by a magnitude and direction, known as the magnitude-angle form.

3. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar only has magnitude. This means that a scalar can be represented by a single number, while a vector requires multiple components to fully describe it. Scalars are used to describe quantities like temperature or speed, while vectors are used to describe quantities like displacement or velocity.

4. How are vectors used in real-world applications?

Vectors are used in many real-world applications, such as navigation systems, 3D modeling, and physics. They are also used in computer graphics, robotics, and engineering to represent and manipulate spatial data.

5. What is the significance of the dot product and cross product in vector calculations?

The dot product and cross product are two important vector operations used in calculations. The dot product is used to find the angle between two vectors and the cross product is used to find a vector that is perpendicular to two given vectors. These operations are important in solving problems involving forces, work, and motion in three-dimensional space.

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