MECHANICS (VECTOR GEOMETRY)

THIRD TERM E-LEARNING NOTE

SUBJECT: FURTHER MATHEMATICS CLASS: SS 2

SCHEME OF WORK

 

WEEK FOUR

TOPIC:MECHANICS (VECTOR GEOMETRY)

SCALAR OR DOT PRODUCT OF TWO VECTORS

The scalar or dot product of two vectors a and b is written as a.b and pronounced as (a dot b). Therefore, a.b =|a| |b| cos dot is defined as a.b = a b cos where is the angle between vectors a and b

If a = a₁ I + a₂j and b = b₁ I b2j

Thus a .b = (a)1bi ii + ab2j I 1 + 2 bi I h +a2 b2 j

Recall that I and j are mutually perpendicular unit vector hence

i.i = |x| cos 0 =1

i.j = |x| cos 90 =0

j.i = |x| cos 90 =0

j.i =|x| cos 0 =1

Hence, a.b =a₁b₁ + a₂ b₂

Examples

Find the scale product of the following vectors 9i -2j + k and I – 3j -4k

Solution:

A=(9i- 2j +k) and b= (i-3j -4k)

a.b = (9i-2j +k) (i-3j-4k)

=9 (1) -2(-3) + 1(-4)=9+6-4a.b =11

2. Let a = 3i+2j, b = -4i+2j and c = i+4j, calculate a.b, a.c and a. (b+c)

Solution:

I a.b = (3i + 2j ) (-4i+2j) = 3 (-4) +2(2)

= -12+4

=-8

II a.c = (3i+2j) (I +4j)

= 3 (1) + 2 (4)

= 3+8 = 11

III a.(b+c)

Find (b+c) = -4i + 2j +i +4j

=-3i +6j

(b+c) = (3i+2j) (-3i +6j)

=3(-3) + 2(6)

= -9+12 = 3.

PERPENDICULARITY OF VECTORS:

If two vectors P and q are in perpendicular directions, thus p.q =0

Example 1: show that the vectors p = 3i+ 2j and q= -2i + 3j are perpendicular.

Solution:

P:q = (3i+2j) (-2i +3j)

=3(-2) + 2(3)

=-6+ =0

Since p.q=0, then the vectors p and q are perpendicular.

2. If p= 4i + kj and q=2i – 3j are perpendicular, find the value of k, where k is a scalar..

Solution:

p.q=0

(4i+kj)(2i-3j)=0

4(2) + k(-3)=0

-3k=-8

K=8/3.

EQUAL VECTORS: Vectors p ad q are equal if p is equal to q.

Example: find the value of the scalar K for which the vectors 2ki + 3j and 8i+kj

Solution:

2ki +3j = 8i +kj

Hence, 2ki =8i, 3j=

2k = 8 12=3k

K=8/2 k=12/3

K=4 k=4

EVALUATION

The vectors AB and C are -2i+6j-3k and -2i-3j+6k respectively. Find the scalar product AB.AC

Find the value of the scalar A for which the pairs of vectors 5i +3j and 2i-4Aj are perpendicular.

ANGLES BETWEEN TWO VECTORS

Is the angle between two vectors and from dot product where a.b=|a|b| cos . Hence, Cos

Where = Magnitude of vector a= ²₁ +²₂

|b| =Magnitude of vector b=²₁ +²₂

Example:

Find the angle between the vectors pp=2i – 2j + k and q=12i +4j – 3k

Solution:

Cos =

p.q= (2i-2j+k)(12i+4j-3k)

=2(12) -2(4)_1(3)

= 24-8-3

=13

|p|=² + (-2)² + 1²=+4+1 ==3

|q|= ² + 4² + (-3) = 144+16+9= 169 =13

Cos

Cos =1/3. =Cos^-1 (1/3)

DIRECTION COSINES A VECTOR:

The direction is specified by the angles which the vector makes wit x and y axes. If we represent these angles by and respectively then,

Cos = Cos =

Example: find the direction cosine of the vector 4i + 3J – 11k

Solution:

Let a = 4i + 3J-11k

|a|= ² + y² +z² = |=² + ² + (-11)² =

Direction cosine, Cos Cos =Cos=

EVALUATION

Find the angle between the vectors 2i + 3j +6k and 3i+4j+12k

Find the direction cosine of vector a = 10i- j+2k

EVALUATION: find the projection of the vector a on the vector b if a=5i-4j+2k and b=6i – j +3k

GENERAL REVISION EVALUATION

Given that a=4i – 2j +k, b=2i – j +3k and c=5i +2k find (i) (a+b)c (ii) a=c+b.c

If a = 4i – 2j +k, b=6i +5j find (i) the unit vector in direction of b. (ii) the projection of a on b (iii) the unit ve4xtor in the direction of a (iv) the projection of b on a.

READING ASSIGNMENT: Read vector Geometry, Further Mathematics project II page 236-240

WEEKEND ASSIGNMENT

If = 3i + 4j and b=gi +2k are perpendicular, what is the value of g? A.-4 B.3 C.-8/3

Find the value of the scalar k for which the vectors ki + 8j and 3i + are equal. A. 3 B.6 C.9

Find the projection of the vector a on the vector if a = 4i + 6j and b=3i-2j. A.-3\ 52 B.5\ 13 C. 0

Calculate the angle between a = -4i +2j and b =I -3j. A.45⁰ B.60⁰ C.135⁰

Find the scalar product of vectors – 2i-3j and 4i +5j? A.-23 B.23 C.7

THEORY

1a) Given that a = 4i – 5j + 2k and b = -7i + 3j – 6k find the scalar product of a and b (b) find the direction cosine 2a + 3b

2 ) Find the angle between p = 6i + 2j – 4k and q = 9i + 5j