Pure Math Part 7 Vector

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2009 A-level Maths Tutor All Rights Reserved www.a-levelmathstutor.com


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This book is under copyright to A-level Maths Tutor. However, it may bedistributed freely provided it is not sold for profit.

Contents

general properties of vectors 3

vector equations 8

the Scalar Product 13



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General Properties of Vectors

Notation

A non-zero vector is has the magnitude of a positive real number and a direction in space.

A vector may be represented by two letters describing a line.The order of the letters

indicates the direction and the length of the line its magnitude.

An alternative to this notation is to use a single bold letter, for example C. Then the

magnitude is |C| or C.

The Unit Vector

A unit vector eg a, has a magnitude of one |a|=1 and can point in any direction.

Sometimes a unit vector is written with an accent over it .

Different unit vectors point in different directions.

Hence, if Fis an ordinary vector,

a unit vector in the direction of F(^ circumflex accent)



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2D representation

P is a point in thex-yplane with coodinates (x,y). iis the unit vector along thex-axis and

jis the unit vector along they-axis.

With Q at (x,0) and R at (0,y):



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3D representation

P is a point inx-y-z space with coodinates (x, y, z). iis the unit vector along thex-axis,jis

the unit vector along they-axis and zis the unit vector along thez-axis.



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Addition(Sum) of vectors

This is also called the Parallelogram or Triangle Law.

If two vectors(a& b) are represented in magnitude and direction by the adjacent sides of

a parallelogramfrom a point, then their resultant(r) is represented in magnitude and

direction by the diagonal of the parallelogram passing through the point.

If two vectors are represented in magnitude and direction by the adjacent sides of a

triangle, taken in order, then their resultant is represented in magnitude but opposite in

direction by the third side.



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Scalar Multiplication

Multiplying a vector by a scalar quantity changes its magnitude but not its direction.



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Vector Equations

Component rules

Consider two vectors:

a=x1i +y1j+ z1k and b=x2 i+ y2j+ z2k

in three dimensional space.

a= b implies that x1=x2 y1=y2 z1=z2

a + b = (x1+x2)i + (y1+y2)j + (z1+z2)k

a - b = (x1-x2)i + (y1-y2)j + (z1-z2)k

ma= mx1i + my1j+ mz1k where m is a scalar quantity

in 3D space, if point A has position vector aand point B has position vector bthen the

distance AB is given by:



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Equation of a straight line - single point & parallel vector given

A(x1y1z1) is a fixed point on the line

ais the position vector for point A a=x1i+ y1j+ z1k

sis a vector parallel to the line s = li+ mj+ nk

( l m nare called the direction ratios of the line)

ris the position vector for an arbitrary point P(x, y, z) on the line.



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Equation of a straight line - two points given

A(x1y1z1) is a fixed point on the line

ais the position vector for point A a=x1i+ y1j+ z1k

B(x2y2z2) is a fixed point on the line

b is the position vector for point B b =x2i+ y2j+ z2k

ris the position vector for an arbitrary point P(x, y, z) on the line.



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Example #1

Given that a= 3i-j+ 2k and b= 2i+j- 2k,

find a.band the included angle between the vectors to 1 d.p.



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Example #2

What is the vector equation describing the straight line passing through the points A(-8, 1,

-2) and B(10, -1, 3)?

Find the coordinates of a point P on AB such that OP is perpendicular to AB(origin O),

hence find the distance OP to 2d.p.



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Notes

This book is under copyright to A-level Maths Tutor. However, it may be

distributed freely provided it is not sold for profit.


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Pure Math Part 7 Vector