3D Vectors Notes- CAPE
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1 S. Kenny-Bennett March 12, 2014 O P 1 A P 2 l m 3D Vectors General Information Given = + + , the point V has coordinates , , and the vector has magnitude = 2 + 2 + 2 + = and so a unit vector in the direction of is = 1 + + . Whereas the direction of a vector in 2 dimension could be determined by looking at the angle the vector makes with the positive x-axis, the direction of a vector in 3 dimension is given by looking at the angle the vector makes with each of the coordinate axes leading to what is known as the direction ratio or the direction cosines of a vector. The Direction Ratio of = + + is : : . Looking at the angle the vector makes with each coordinate axis the Cosine Ratios are developed cos = = cos = = cos = = The cosine ratios lead to 2 additional relationships 1) 2 + 2 + 2 = 1 2) = , = , = The Equation of a Line The idea of the equation of a line in the 3 dimensional space is tied to the concept of locus of a point satisfying certain conditions. Using a vector parallel to the line and the position vector of a point on the line an equation/formula can be developed which yields the position vector of any point P on the line. Given the vector of a line l in vector form let = + + . We have the task of writing an equation to represent m a line passing through A 1 , 1 , 1 parallel to l. Since m is parallel to l the vector representing m is a scalar multiple of that representing l. As such we can say that = . Selecting a random point P on m we can also say that = since is parallel to l. Using the triangle law of vectors and the position vector of A, , we can say that = + ⇒ = + . is the vector that describes movement from the origin to any point P with coordinates , , just as the point M with coordinates, would be any random point on the line = + once the coordinates of M satisfy the equation of the line. Now = 1 + 1 + 1 + + + represents the vector equation of the line m the various points P will be on the line m and so we have the locus of the set of points, P, in a specific direction from the point A which is on m. If = 1 + 1 + 1 + + + is the vector equation of a line, then from this form we can write parametric equations for x, y, and z with parameter λ. By making λ the subject of the equations we can equate them and create the Cartesian equation of the line. Parametric Equations Make λ the subject of each equation equate to write the Cartesian equation = 1 + = 1 + = 1 + = − 1 = − 1 = − 1 − 1 = − 1 = − 1 = Notice that in each form of the equation the direction ratio : : is present.
description
Advanced level math notes on 3D vectors
Transcript of 3D Vectors Notes- CAPE
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S. Kenny-Bennett March 12, 2014
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3D Vectors
General Information Given = + + , the point V has coordinates , , and the vector has magnitude =
2 + 2 + 2+
= and so a unit vector in the direction of is =1
+ + .
Whereas the direction of a vector in 2 dimension could be determined by looking at the angle the vector
makes with the positive x-axis, the direction of a vector in 3 dimension is given by looking at the angle
the vector makes with each of the coordinate axes leading to what is known as the direction ratio or the
direction cosines of a vector.
The Direction Ratio of = + + is : : . Looking at the angle the vector makes with each coordinate axis the Cosine Ratios are developed
cos =
=
cos =
=
cos =
=
The cosine ratios lead to 2 additional relationships 1) 2 + 2 + 2 = 1
2) = , = , =
The Equation of a Line The idea of the equation of a line in the 3 dimensional space is tied to the concept of locus of a point
satisfying certain conditions. Using a vector parallel to the line and the position vector of a point on the
line an equation/formula can be developed which yields the position vector of any point P on the line.
Given the vector of a line l in vector form let = + + . We have the task of writing an equation to represent m a line
passing through A 1, 1, 1 parallel to l. Since m is parallel to l the vector representing m is a scalar multiple of that
representing l. As such we can say that = . Selecting a
random point P on m we can also say that = since is parallel to l. Using the triangle law of vectors and the position
vector of A, , we can say that = + =
+ . is the vector that describes movement from the origin to any point P with coordinates , , just as the point M with coordinates , would be any random point on the line = + once the coordinates of M satisfy the equation of the line.
Now = 1 + 1 + 1 + + + represents the vector equation of the line m the various points P will be on the line m and so we have the locus of the set of points, P, in a specific direction from
the point A which is on m.
If = 1 + 1 + 1 + + + is the vector equation of a line, then from this form we can write parametric equations for x, y, and z with parameter . By making the subject of the equations we
can equate them and create the Cartesian equation of the line.
Parametric Equations Make the subject of each equation equate to write the Cartesian equation
= 1 + = 1 + = 1 +
=1
=1
=1
1
=
1
=
1
=
Notice that in each form of the equation the direction ratio : : is present.
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Relationships between Lines Given 2 lines = + + + 1 + 1 + 1 and = + + + 2 + 2 + 2 , we can classify the lines as parallel, intersecting or skew lines based on the direction ratio and whether or
not there is a point of intersection. Two lines are parallel if their direction ratios are the same. Otherwise
the two lines are expected to intersect at one point since they are not parallel. However, if the lines are in
different planes which are parallel to each other then the lines are classified as skew lines since they are
neither parallel nor intersecting.
To determine whether the lines are parallel; compare their direction ratios. Is 1: 1: 1 = 2: 2: 2? If the direction ratios are different then try to find out if there is a common point for the lines. This is done
by solving the simultaneous equation of the parametric equations of the components for that will satisfy all three equations at the same time.
= + 1 = + 2 = + 1 = + 2 = + 1 = + 2
If a value for that satisfies all three equations cannot be found the lines should be classified as non-intersecting or skew lines.
The Angle between 2 lines The angle between 2 lines is the angle between their direction vectors. To determine the angle, use the
same formula for angle between 2 vectors ( i.e. cos =
).
Using the unit vector in the direction of the direction vectors of the respective lines may prove to be
useful as the product of the magnitudes would be 1.
The Equation of a Plane The idea of the equation of a plane in the 3 dimensional space is also tied to the concept of locus of a
point satisfying certain conditions just as in the case of the equation of the line. Using a vector
perpendicular to the plane and the position vector of a point on the plane, the fact that the scalar product
of perpendicular vectors is zero, is manipulated to create the vector equation of the plane, sometimes
referred to as the scalar equation of the plane. Planes are usually denoted using the symbol (capital ).
Some other important things to note
1) = 1 the dot product of a unit vector and itself is 1 2) = the dot product is commutative
3) = the dot product is distributive
Since n is perpendicular to , then the dot product of any vector on and n is zero.
I.e. = 0 and = 0 since and are perpendicular to .
and will also be perpendicular to a unit vector in the direction of n.
NB. If = then =
Since = = we have that = 0
Solve the system of equations by solving two equations first and
then testing the solutions obtained for and to see if the values
satisfy all three equations.
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Rewriting = 0
= 0
( ) = 0
1 = 0 =
Repeating the process with a we get that = Equating we have that = = It follows that we can represent this relationship between r, a
and n by multiplying both sides of the equation by d to get
=
NB
The Vector Equation of the plane , is = where
= 1 + 1 + 1 is the position vector to the point 1, 1, 1 on the plane
= + + is a vector perpendicular to the plane
= + + is the position vector to any point , , on = gives the distance from the origin to the plane and so = = can be used to
determine the distance of the plane from the origin by dividing D by the magnitude of .
I.e.
= , the distance of the plane from the origin.
The Cartesian Equation of the plane is + + =
E.g. Let a plane l be perpendicular to the vector = 2 + 3 4. Let the point A 2, 1, 3 be on the plane l.
a) Write the vector equation of the plane l. b) Hence write the Cartesian equation of l.
Since the plane l is perpendicular to = 2 + 3 4 then the dot product of any vector on the plane l and
= 2 + 3 4 is zero. Let R , , be another point on the plane l and N be the point where = 2 + 3
4 touches the plane l.
So and are vectors on l that are both perpendicular to = 2 + 3 4 and their dot products are equal.
However, we do not have components for and but we can find them using the Triangle Law of vectors as
shown above. From the working above we know that = and = and by multiplying by d we get
= .
= 2 2 + 3 1 + 4 3 = 4 + 3 12 = 5
The vector equation of the plane l is therefore = 5 2 + 3 4 = 5
Since = + + the Cartesian equation is 2 + 3 4 = 5
NB the plane l is
= =
5
29 away from the origin in the direction of = 2 + 3 4.
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