Vectors 2

Vector Recap

 

Direction Ratios and Cosines

 

Example

Show that the points A(-1,-8,-2) ,
B(2, -5, 4) and C(3, -4,6)  are collinear, and
establish the direction cosines of AB

 

Vector Product

The  vector product a x b  of 3D vectors
a and b is a vector perpendicular to the plane
containing a and b.

 

Since the resulting vector is a normal to the plane,
it is often called n and has magnitude 

This gives

 

The direction of n is found using the
Right Hand Screw rule.

If you curl your hand from a to b ,
your thumb will point towards n.

Looking at the unit vectors i,j,k

 also,

  

Other properties

  

Components

Vectors a and b can be written in unit vector form
as 

 

which is why it is also know as the cross product.

 

To find a unit vector perpendicular to vectors
a and b , calculate   

Example

Evaluate

Note

 

Example

 Calculate the distance from A (3,2,1) to the
straight line passing through B (4,5,6) and C ( 7,8,9)

 

  

Scalar Triple Product

This describes the volume of a parallelepiped
which has edges a, b and c.

 

The base   is  separated by angle θ

 

Example

Find the volume of the parallelepiped
with vectors
  a = i +2j -3k    
  b = 2i -2j + k    
 c = i - 4k    

Planes

 

Let P(x,y,z) be a general point on plane π,

 i

s a normal to π passing through point A.

 

Steps:

• Identify the normal.
  (Use the cross product if necessary)
• The dot product of the normal and each point on the plane will be equal.
• Find the dot  product of the normal and one given point on the plane.
• Use the result to find the equation of the plane.

 

Example

Find the equation of the plane perpendicular to
the vector n = 2i + 3j + 6k at the point G( 1,2,0).
Does the point T(2,4,0) lie on the plane ?

  

 

Example

Given R (1,2,3) and S ( -2,-1,-3), find the equation
 of the plane passing through R which is perpendicular
to RS.

 

Example

Find the equation  of the plane passing through
A (1,3,1), B (2,0,-2) and C ( 4,5,6)

   

 

A plane in space can be found if

 

• 3 points on the plane are known

 

 

• 2 non-parallel intersecting lines on the plane are known

 

• One point on the plane and a normal to the plane are known

 

Vector Equations