How Do You Formulate the Vector Equation for a Line Given a Point and Direction?

In summary, the conversation is about finding the vector equation of a line that passes through a given point and is parallel to another line passing through another given point. The attempt at a solution involves using the point-slope form and determining the equation using the given points. However, since there are an infinite number of possible lines that can pass through the second point, the answer cannot be determined without further information.
  • #1
styxrihocc
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Homework Statement



Find the vector equation of a line that passes through the point (1,3,4) and is parallel to a line that passes through point (5,1,2)

Homework Equations


The Attempt at a Solution


<1,3,4> + t<5,1,2>
<1,3,4> + <5t,t,2t>
<1+5t,3+t,4+2t>

Is my answer correct? if not please point me to the right direction since I'm not quite sure how to solve this problem, thank you.
 
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  • #2
styxrihocc said:

Homework Statement



Find the vector equation of a line that passes through the point (1,3,4) and is parallel to a line that passes through point (5,1,2)
Is this the problem exactly as given to you? If so, you can't answer it because there are an infinite number of lines which pass through the point (5, 1, 2).
 

FAQ: How Do You Formulate the Vector Equation for a Line Given a Point and Direction?

What is a vector equation of a line?

A vector equation of a line is an equation that represents a line in three-dimensional space using a vector and a point on the line. It is typically written in the form of r = a + tb, where a is the position vector of a point on the line and b is the direction vector of the line.

How is a vector equation of a line different from a parametric equation of a line?

A vector equation of a line and a parametric equation of a line both represent a line in three-dimensional space, but they differ in their forms. A vector equation uses a vector and a point on the line, while a parametric equation uses two or more parameters to represent the coordinates of points on the line.

What information does a vector equation of a line provide?

A vector equation of a line provides information about the direction and position of a line in three-dimensional space. The direction vector represents the slope of the line, while the position vector represents a point on the line.

How do you find the direction vector of a line from its vector equation?

The direction vector of a line can be found by isolating the variable t in the vector equation and then expressing the equation in the form of t = k, where k is a scalar. The coefficients of the remaining variables will give the direction vector of the line.

Can a line be represented by more than one vector equation?

Yes, a line in three-dimensional space can be represented by infinitely many vector equations. This is because each point on the line can be represented by a different position vector, resulting in a different vector equation.

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