pure math part 7 vector
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Contents
general properties of vectors 3
vector equations 8
the Scalar Product 13
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3
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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