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 = aI + a2j and b = bI 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 =a1b+ ab2

Examples

1.                   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

a.                   (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

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

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

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