1. Vectors

2. What you need to know
How to add and subtract vectors and multiply by a scalar
How to find the magnitude of a vector
How to calculate the scalar product and how it relates to angles between vectors
How to use vectors for co-ordinate geometry
How to use the vector eqn of lines in the form r = a +tb
How to find the intersection of lines, or show that they are skew lines (non-parallel and non-intersecting)
How to find the angle between lines

3. The Length of a 3D Vector Find the length of the vector 3i + 5j - 2k
= √ ( -2 ) 2
= √ 38

4. The vector equation of a straight line passing through the point A with position vector a, and parallel to the vector u R u
R = a general point on the line A
OR = OA + AR r
r = a + λu a O
where λ is a scalar parameter

5. The vector equation of a straight line passing through the points A and B, with position vectors a and b. R
AB = b - a
R = a general point on the line B
OR = OA + AR A
OR = OA + λAB b a
r = a + λ( b – a ) O
where λ is a scalar parameter

6. Vector Equation of a Line in 3D Example 1
Find the equation of the line passing through the point ( 3, 4, 1 ) and ( 5, 7, 7 )
Equation is given by
r = a + λb
r = 2 3 6 ( ) 4 1
+ λ

7. The Angle Between Two Vectorsθ b
scalar product ( dot product )
cos θ = a . b
|a| |b|
Note: If two vectors a and b are perpendicular
then a . b = 0

8. The Angle Between Two Directions Example 2
Find the angle between the following pair of vectors:
a = i + 4j + 2k and b = 2i + 5j + 7k
|a| = √ ( ) = √ 21
|b| = √ ( )= √ 78
a . b =
1 x x x 7 = 36
cos θ = a . b
|a| |b|
cos θ =
√21 √78
θ = 27.2°

9. The Angle Between Two Lines in 3D Example 3
r = 2 -1 ( ) 1 4
+ λ
r = 3 1 ( ) 2 -1
+ μ
The angle between the lines is the angle between their directions
cos θ = a . b
|a| |b|
cos θ = x (-1) x (-1) x 1
√2 2 + (-1) 2 + (-1) 2 √
cos θ =
√ 6 √ 10
θ = 49.8°

10. 4. Past paper question Jan09 no7
Show that the straight line with equation meets the line passing through (9, 7, 5) and (7, 8, 2) and find the point of intersection.
2nd line eqn is
Intersect when

11. 4. Past paper question Jan09 no7
So lines intersect at (5, 9, -1)

12. Past paper question Jan08 no5
 Example 5 The vector equations of two lines are
and
Prove that the two lines are perpendicular and skew

13. Past paper question Jan08 no5
 Use the direction vectors only
and
Lines are perpendicular when dot product =0

14. Past paper question Jan08 no5
 would cross
when
Solving top two eqns gives

15. Past paper question Jan08 no5
 Substituting into equations gives
So lines do not intersect
They are not parallel (seen in part (i))
So they are skew lines
You have to have both bits here

16. Additional example
C is the point (3,12,3) and the point P is on the line CP is perpendicular to the line. Find the co-ordinates of P

17. CP is perpendicular to the line –
We need scalar product with the direction vector