IB Math SL Year 2
Name: ____________________________
Date: ______________
Lesson 6-6: Manipulating Vectors
Today’s Goals: 1. How do I determine a vector given its coordinates? What is a unit vector, how do I get it?
Let’s start basic…
How far did vector a translate (solid to dotted)? In what direction(s)?
Graphically
Algebraically
How did you
get it?
Now, apply it…
Consider points A and B, where A(–1, 2) and B(4, 8).
1. Sketch the two points, on the coordinate plane and write the position vector of AB?
2. How can we obtain this same position vector algebraically using our original points?
Let’s Try this again!
3. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).
Together: Find the vectors OB and AC .
Flashback! What is unit vector?
⃑⃑⃑⃑⃑ and 𝐴𝐵
⃑⃑⃑⃑⃑
Your turn: Find the vectors 𝑂𝐴
IB Math SL Year 2
̂ ) of vector a.
With that, what would sketch the unit vector (𝒂
How did we get this?
̂=
To get a unit vector in the same direction of a given vector: 𝒗
𝟏
|𝒗|
𝒗
Let’s apply it…


 




The vectors i , j are unit vectors, and the vectors u = – i + 2 j and v = 3 i + 5 j are given.
(a)




Find u + 2 v in terms of i and j .



A vector w has the same direction as u + 2 v , and has a magnitude of 26.
(b)
Practice



Find w in terms of i and j .
IB Math SL Year 2
̂, that is in the same direction as p.
1. With vector p = 2i + 5j, write the unit vector, 𝐩
2. The following diagram shows the point O with coordinates (0, 0), the
point A with position vector a = 12i + 5j, and the point B with position
vector b = 6i + 8j. The angle between (OA) and (OB) is .
y
C
Diagram not to scale
B
A
Find
(i)
| a |;
O
(ii)
(iii)
a unit vector in the direction of b;
the exact value of cos . in the form
p
, where, p, q ∈ .
q
x
IB Math SL Year 2
3. Three of the coordinates of the parallelogram STUV are S(–2, –2), T(7, 7), U(5, 15).
(a)
Find the vector ST
(b)
⃑⃑⃑⃑
Find the vector SU
(c)
⃑⃑⃑⃑ .
Find the unit vector u in the direction of SU
4. The position vector of point A is 2i + 3 j + k and the position vector of point B is 4i − 5 j + 21k.
(a)
(i)
Show that AB = 2i −8 j + 20k.
(ii)
Find the unit vector u in the direction of AB .
IB Math SL Year 2
7
10 
5. The diagram shows a parallelogram OPQR in which OP =   , OQ =  .
 3
1
y
P
Q
O
x
R
(a)
Find the vector OR .
(b)
Use the scalar product of two vectors to show that cos( OP̂Q )= –
15
754
.
IB Math SL Year 2
6. The diagram below shows a cuboid (rectangular solid) OJKLMNPQ. The vertex O is (0, 0, 0), J is (6, 0, 0),
K is (6, 0, 10), M is (0, 7, 0) and Q is (0, 7, 10).
(a)
(iii)
(i)
  6
 
Show that JQ =  7  .
 10 
 
Find MK .
The point O has coordinates (0, 0, 0), point A has coordinates (1, –2, 3) and point B has coordinates (–3, 4, 2).
(a)
(i)
  4
 
Show that AB =  6 .
 1
 
(ii)
̂ O, in radians.
Find angle BA
(iii)
Find the unit vector u in the direction of AB .