Vectors and Planes

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Introduction to Vectors and Scalars

DEFINITION: Vectors were developed to provide a compact way of dealing with multidimensional situations without writing every bit of information. Vectors are quantities that have magnitude and direction, they can be denoted in three ways: in bold (r), underlined (r) or A->B.

The position point of a vector is defined using Cartesian co-ordinates: it uses the coordinates of the OX, OY and OZ axes where O is the origin. We will be looking at vectors in 3 dimensional space in Cartesian coordinates. Similar ideas hold for vectors in n dimensional space (n-vectors).

graph1

In the above diagram r is the position vector of a point A relative to the origin O. So matixA.

Try plotting your own vectors.

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Addition and Subtraction of vectors

DEFINITION: We will look at two laws involving addition and subtraction; The commutative law and the Associative law.

The commutative law:

Addition and subtraction of vectors obeys the commutative law,

matixA matixA

This means that:

matixA

In terms of the following components

components1

For addition:

components2 lines

For subtraction:

components3 lines

The Associative law:

This uses different routes to get to the same final desination

graph2 graph2

In terms of the following components:

components4

components5

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The length and unit vector of a vector

The length of a vector :

DEFINITION: For any vector like this one

components

The length of a (which is denoted by |a|) is given by:

a

The unit vector of a vector

DEFINITION: A unit vector is a vector with unit length 1. By standard convention we let i, j and k be unit vectors along the positive x, y and z axes, so in terms of components:

a

a

Find the length of your own vector and the related unit vector.

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Scalar multiplication of vectors

DEFINITION: This involves the multiplying of a vector by a scalar, i.e. a number.

In terms of the following components:

components

components

Scalar multiplication will change the length of the vector and if the factor λ is negative the vector will point to the opposite direction. Scalar multiplication satisfies the following properties where a and b are vectors, λ and μ and are scalars.

components

Now try multiplying your own vectors and scalars.

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The vector equation of a line

DEFINITION: As taught in A-level and before the equation of a straight line is given by y = mx + c where m is the gradient and c is the y-intercept of the line. This is for two dimensions. For three dimensions similar concept can be used to represent a line in vector form.

a

In the diagram above, the vector (1,m) is parallel to the line AB and point A with vector coordinates (0,c) lies on the line AB. Let B be a typical point on the line with positive vector r. As d=(0,c) is a point on the line and n=(1,m) is a vector parallel to the line, the vector equation of the line AB is given by

a, a.

Find the vector equation of your own line by entering two points.

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Intersections of vectors with the X-Y plane

Continuing from above we will now look a case where a given line intersects the X-Y plane. The vector equation of the line is

a

If we take:

a

we get that

a

We know that in the x-y plane z = 0 so;

a

Substituting this into equations will give that λ = -2, which when substituted into the second and first equation give that y = -2 and x = -2. The intersection of the above line occurs at the point(-2,-2,0).

Look at another example.

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Scalar product

Let us take two vectors.

Vectors

Then the scalar product of a and b, denoted by a.b is

scalar product

Setting these equal to each other gives

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The angle between two vectors

Using the two above formulae and setting them equal to each other as shown below we are able to calculate the angle between two vectors.

a

Try finding the angle between vectors yourself using the scalar product.

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Vector equations of planes

DEFINITION: Let a vector normal (vector perpendicular to the plane) be denoted as n, a position vector of some point in plane denoted as a and a typical point on the plane denoted as r.

a

Looking the above figure, ra is perpendicular to n. So the formula of the vector equation of the plane is given by:

a

Try finding the vector equation of your own planes.

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Angles of intersection of two planes

DEFINITION: The angle between two planes is the angle between the two normals. The plane must firstly been written in vector form.

For example let our plane be a

In vector form this is

a

where

a

this can also be written as

a

If we had two planes then we would have two normal vectors say n1 and n2. to find the angle between these two vectors using the same formula when we found the angle between vectors (above).

Find the angle between two of your own planes.

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Vector Product

DEFINITION: The vector product is fundamentally different from the scalar product. The vector product of two vectors is a vector but the scalar product is a scalar. The vector product is given by:

a

where
|a| is the length of a
θ is the angle between vectors
n is the unit vector perpendicular to a and b whose direction is determined by the left hand skew rule.

a

For vectors a × b is found by using the following:

a

For simplicity this can be written in terms of determinants.

a

Now try the vector product yourself.

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The Area of a Vector Triangle

DEFINITION:In terms of vectors the area of the triangle below is:

a

Try finding the area of your own vector triangles.

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Finding the Equation of a Plane given Three Points

Recall that the vector equation of a plane is (r - a). n = 0 where a is a point on the plane and n is a vector normal to the plane.

Suppose we have the points

components4

which are in Cartesian form. We firstly need to find the vector parallel to the plane.

a

To get a vector n, which is normal to the plane, we take the vector product of the above vectors.

a

This gives a vector denoted by n. So the equation of the plane is found using the same method as above.

By substituting in we get

a

Find the equation of your plane given three of your own points.

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Minimum distance between two skew lines

DEFINITION: Skew lines are lines or vectors which are not parallel and do not meet. We now seek the minimum distance between these lines. By drawing a line between both lines, called a transversal, it will be perpendicular to both lines.

a

The transversal connects A and B, n3 is the unit vector in the direction AB and p is the required distance. As mentioned above n3 is perpendicular to both n1 and n2.

a

Find the minimum distance between your own lines.

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