1. LECTURE 5 OF 8
Topic 5: VECTORS
5.5 Application of
Vectors In Geometry

2. LEARNING OUTCOMES
find a vector equation of the straight line
in the form of
where t is a scalar.
(b) find the parametric and cartesian
equations of a straight line.

3. The Equation of a Line In Space
A line in space is a straight line. Suppose
that R (x,y,z) is a point which is free to move
on a line containing a fixed point A (a1,a 2,a3). If
is a direction vector of
the line
NOTE :

4. , t is a scalar
A vector equation of a line is :
NOTE : To find a vector equation of a line, we have to determine a position vector of one point on the line and any vector which is parallel to the line, then use the above equation .

5. In the component form, it can be written as
Equating the respective components,
we obtain,
x = a1 + tv1
y = a2 + tv2
z = a3 + tv3
The parametric equations of the line

6. x = a1 + tv1
y = a2 + tv2
z = a3 + tv3
tv1 = x - a1
tv2 = y – a2
tv3 = z – a3
Isolating t in each of these equations gives,
The Cartesian equations of the line

7. Find the vector equation,the parametric
Example 1
Find the vector equation,the parametric
equations and the Cartesian equation of
the line through (1,-2,3) in the direction of
Solution
the vector equation is

8. The parametric equations are
x = 1 + 4t,
y = t,
z = 3 - 6t
The Cartesian equations are

9. Example 2
Find a vector equation for a line passing through a point with position vector and is parallel to
the line with cartesian equation
Hence find a point on the line when y = 0

10. Solution The vector equation is
A general point on the line has coordinates
x = 5 + 3t, y = t, z = t
If y = 0
0 = t
t =

11. when t =
The point on the line where y = 0 is

12. Example 3
a) Find parametric equations for the
line L passing through the points
A (2,4,-1) and B (5,0,7).
b) Where does the line intersect the
xy-plane ?

13. Solution a) The vector equation is The parametric equations
x = 2 + 3t, y = 4 - 4t, z = t

14. b) The line intersects the xy-plane
at the point where z = 0
So, when z = 0
0 = t
t =
When t =
x = 2 + 3( )
= 2 + =

15. y = 4 - 4( )
= 4 - =
The line intersects the xy-plane at

16. Conclusion
the vector equation of the line :
To find the equation, determine

17. The vector equation of the line :
The parametric equations of the line :
x = a1 + tv1
y = a2 + tv2
z = a3 + tv3
The Cartesian equations of the line