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# In this task, we are going to show how any two vectors are at right angles to each other by using patterns with vectors.

Extracts from this document...

Introduction

Introduction

In this task, we are going to show how any two vectors are at right angles to each other by using patterns with vectors.

This will be achieved by firstly plotting two vectors and discussing any similarities and difference between them. Then two randomly selected points will be chosen and a line drawn through them. Different ways of representing this vector will be explained. Finally, the general form of a vector equation will be used to determine and prove how two vectors are perpendicular to each other.

The use of the Geogebra computer software will be used to graph each vector although the methods used will be explained.

Vectors are used to display the magnitude and direction for a path of an object. Examples include velocity, acceleration, force, displayment, momentum and weight. A vector can be written in 3 different forms:

Velocity vector:

Parametric equation:  x= a+ct           y=b+dt

Cartesian equation:

Vector can be drawn with an initial point (a, b) and then a direction vector . The line that goes through the set of points has an arrow at the end to show its direction.

Middle

,

,

t = 3,      ,

,

t = 4,      ,

Again from these points for t it is now possible to plot them on a set of Cartesian axes. This was done on the Geogbra computer program. It follows the idea of simply points on the x, y axes when based at the origin.

Another graph can be created by plotting those vectors  with the equation. For  .

By comparing those two equations, we are able to see there is a pattern that although the value of (t) changes, the vectors  are collinear. This means they lie on a same straight line with the same slope.

In the first equation, the slope of it is lying on the positive side, but the second slope is lying on the negative side. It is considering to their vectors direction  and

In this time, we chose the point and  from the vector equation:. Where r = 1, r = 3. After that, we draw the line L that those two point can pass through it.

figure 3

The slope of these two point is same as the slope of the equation .  (Refer to figure 2)

We assume that the vector  is  so

= td {distance = time × speed}

Conclusion

t is a scalar multiple of arbitrary value, any value in the set of all real numbers.  is a direction vector describing the lines direction. It and its multiples are being vector added to the constant, which is a vector describing the point  which the vector passes through.

If a line L passes through a point U (h, k) in a direction d. Then we can create a vector equation which is thegeneral position vector for point R on line L.

By looking to the diagram,

Figure 7

Now, changing the velocity equationin the form

. Line L passed through a point U (h, k) in a direction

Suppose a plane in space has normal vector

And that is includes the fixed point . U( u, k) is any other point in the plane.

Now  is perpendicular to n

where the RHS is a constant.

could also be written as , which implies.

If a plane has normal vector  and pass through

then is has equation -bu + ak = -bu1 + ak1 = d,     where d is a constant. This is called the Cartesian equation of the plane.

Now, we are trying to subinto, and then provide the thoery above.

where  t = 1

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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