FURTHER MATHEMATICS SS 1 SECOND TERM
WEEK 5 DATE(S)……………………………………….
SUBJECT: FURTHER MATHEMATICS
TOPIC: VECTOR 2
- Scalar multiplication of vectors
- Unit vector
- Direction cosines
- Scalar (dot) product: Application of scalar (dot) product.
- Projections of vectors
- Application of scalar product
Sub-Topic: VECTOR MULTIPLICATION BY A SCALAR/UNIT VECTOR
Let say and m are scalars then the following laws are true for any vectors a and b:
a is also a vector
(a+ b) = a + b (distributive)
( + m)a = a + ma.
If k is a scalar, then ka is a vector which is parallel to a but k times the magnitude of a. If k>0 then ka is in the same direction of a.
However, if k<0, then ka is in a direction opposite to a
THE UNIT VECTOR
The unit vector is an important concept in the study of vectors.
The definition of unit vector was given earlier as a vector which has an absolute value of unity
We now amplify on this concept.
If OP = ai + bj, then we represent a unit vector in the direction of p by . since a unit vector has a magnitude equal to unity
i.e and ll = 1
ll = .
Thus a unit vector in the direction of is given by
Example 1: Given that , , . Find a unit vector in the direction of and write down a vector parallel to
Unit vector in the direction of =
Example 2: Given that , , , where and are scalars, find an. If + .
- Given that the vectors and are parallel, find the constant
- Given that and , find the unit vector in the direction of
Sub-Topic: PROJECTION OR RESOLUTION OF VECTORS
If the position vector of the point A relative to a reference point o is a and that of the point B relative to O is b, we call the length ON the projection of the vector b on the vector a. If a denotes the unit vector in the direction of the vector a then
a =/ậ/ and /ậ/ = 1
/on/ = /OB/ cos Ө
= /b/ cos Ө
=/ȃ/ /b/ cos Ө
Then the projection of the vector b on a is ȃ •b where ȃ is the unit vector in the direction of the vector a.
Also the projection of the vector a on b is a.Where is the unit vector in the direction of the vector b.
As the resolved part of a in the direction of b the projection of the vector a on b can also be viewed.
Examples: Find the projection of the vector p on the vector qif:
- Let the projection of q on p be u, then
/ P/ =
p.q = (2)(-1) + (3)(4) = -3 + 12 = 9
U1 = =
- Let the projection q on p be, U2, then
/p/ = =
p.q = (4)(-1) + (-5)(1) = -9
U2 = X =
- Let the projection q on p be U3, then
/P/ = =
p.q = (6)(2) + (2)(3) = 12 + 6 = 18
U3 = X =
- Let the projection of q and p be U4, then
/p/ = =
p.q = (3)(1) + (2)(3) = 3+6 = 9
U4 = X =
For each pair of the following, find the projection of the second vector on the first vector:
- a = 3 – 4 and b = 5 + 8
- m = 7+ 3 and n = 2–5
- x = 4- 5 andy = 3 – 7
SUB TOPIC:- SCALAR (DOT) PRODUCT/DIRECTION COSINES
The scalar product of two vectors a and b is written a.b and is defined as the product of the lengths of the two vectors and the cosine of the angle between them.
Thus if Ө is the angle between the vectors a and b then a.b =/a/
Let and be two unit vectors which are perpendicular to each other.
a= ax + ay
b=bx + by
a.b = (ax + ay) . (bx + by)
= axbx . + axby. + aybx. + ayby .
As and are unit vectors which are mutually perpendicular to each other we have
- i. i = 1X1 cos00 = 1
- j.j = 1 X 1 cos00 = 1
- i.j =1 x 1 cos 900 =0
- j.i = 1 x 1 cos 900 = 0
So a.b =axbx + ayby
The scalar product can be sometimes be called the dot product.
The angle between two vectors Ө can be defined from the definition of scalar product.
Thus from a.b = /a/ /b/ cosӨ , we have
Cos Ө =
If a and b are parallel then Ө = 0 and a.b =/a/ /b/
If a and are perpendicular then Ө = and a.b =0
- Find the dot product of the following pairs of vectors:
- a = + 4 and b = 5 -3
- p = 6 – and q = 2 – 8
- a.b = ( + 4j) . (5 – 3
=(1)(5) + (4)(-3) = 7
- p.q = (6i – j ) . (2 – 8 )
= (6)(2) + (-1)(-8) = 20
- Find the cosine of the angle between the following pairs of vectors;
- m = 4 + 3 and n= 2 + 5
- S = 7 – 4 and t = 3 – 2
- Let the angle between the vectors m and n be 𝝰1 then
mn = (4)(2) + (3)(5) = 23
Cos𝝰1= = X = =
(b). Let the angle between the vectors s and t be then
/s/ = =
/t/ = =
s.t = (7)(3) + (-4)(-2) = 29
Cos𝝰2 = = = X = =
PROPERTIES OF A SCALAR (DOT) PRODUCT
- Commutative property
Let a = ax + ay , b = bx + by
- Distribution property
- if //. In particular 2//2
- if then
- Multiplication by a scalar (𝛌
- If ⍬ is the angle between and then
Condition for parallelism If the vector x+ ay is parallel to the vector
b =bx + by ,
i.e =𝛌is a scalar) then = = 𝛌
Condition for perpendicularity
If b, i.e then xx + yy = 0
- Given that /p / =3 ,/q/ =4 and =-6, find the angle between p and q (WAEC)
⍬ = cos-1 -0.5 = 1200.
- If = 3- 4 and b = 6 – 8 find the scalar product of and
The direction cosines of vector are:
Where α, β and λ are angles which OR makes with OX, OY and OZ axes respectively, where O is the origin and X,Y and Z are mutually perpendicular directions in a 3 – dimensional plane.
- Given that:
- Find the direction cosines of vectors: (i) (ii)
SUB – TOPIC: APPLICATION OF SCALAR PRODUCT
Example 1: The vertices A,B and C of a triangle have position vectors and respectively , relative to the origin .prove that .
From the triangle above, taking A is the origin and applying dot product we get
By triangle law, a = b – c,
and a.a = (b – c).(b – c)
Substituting for , we get
Example 2: A particle moves from a point with position vector to a point with position vector. A constant force, among other forces acting on the particle is responsible for the movement. Find the work done by the force.
Solution: work done is defined as the product of force and displacement, and it is a scalar quantity.
work done by , where refers to the displacement of the particle.
Work done by
= 10 + 24 = 34joules
- Find the unit vector direction of vectors: (a) (b)
- Find modulus of each of the following vectors :(i) (ii) (iii)
- Find the dot product of the following pairs of vectors:(i)
- If U,V are any two vectors, prove that . Give a geometrical interpretation of this result when: (i) (ii)
- For what values of λ are the vectors and perpendicular
- Find the values of µ and λ so that the forces may be in equilibrium.
- For the following pair of vectors, find the projection of the second vector on the first vector: a = 3 – 4 and b = 5 + 8
- What is the unit vector which is parallel to the vector ?
- If m = 3 – 4 and n = 5 + 8 what is the cosine of the angle between m and n ?
- Given that a= -3 + 4 and b = 2 -, evaluate
- SCALAR/DOT PRODUCT
- UNIT VECTOR
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