Vectors are used as visual and mathematical tools to show the size and direction of displacement, velocity and acceleration in our three-dimensional world. For example to specify the displacement (a change in position) of an object, we need to know how far it has moved along each of three directions: up/down, back/forth, and left/right. Knowing the size (or magnitude) and direction of the displacement provides this information.
Let's start our discussion of vectors by imagining a that spectator travels from Karuizawa, where she was watching a curling match, to Hakuba Village, to watch the downhill event. We can represent her vector displacement for this journey by an arrow with its tail at Karuizawa (her starting position) and its head at Hakuba Village (her final position).
The magnitude of her displacement is just "as the crow flies", the straight line distance between Karuizawa and Hakuba, about 110 km, and the direction of the displacement is seen as horizontal and vertical.
To find the magnitude of the displacement we use the map scale. Earlier, we showed you the equation for calculating the scale factor: (true length)/(the length in the image). The scale for a map relates the distance between two points as measured on the map itself, with the actual distance on the ground. It is often expressed directly, for example "1 cm equals 50 km", or in the form of a ratio, as in "1:5,000,000”. That is, 1 cm on the map represents 5,000,000 cm or 50 km on the earth. Sometimes however, as in the map on the screen, we have to determine the scale by measuring a line in the legend of the map that represents a known distance. Here, for example, you have to use a segment in the legend on the screen (or on the print-out, if you've printed this page) that represents 10 km.
We know that maps indicate direction by north, south, east and west. (They also indicate vertical position by contour lines). Our athlete could, if she wanted, complete her Karuizawa-Hakuba journey by traveling first west 55 km and then north 75 km (or first north and then west). Giving the westward and the northward parts of the displacement uniquely specifies the horizontal vector displacement; these two quantities are the east/west and north/south components of the vector.
This should give you an idea how we begin to quantify an object’s movement in three dimensional space. In linear kinematics, we don’t use map coordinates to describe an object’s movement. Instead of north/south, east/west and up/down we often describe the components using a Cartesian (X-Y-Z) coordinate system. So, we substitute X for east/west, Y for north/south and Z for up/down. Notice the x, y, and z coordinates on the previous graphics.
This is a bit more universal and we don’t have to worry about using a compass. In the vector lesson, we saw how moving in two and three dimensions may be represented by vectors. In the above examples, displacement vectors were used to describe motion that is not just in a straight line. We may also use vectors to denote velocities and accelerations.
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© April, 1998, Montana State University-Bozeman
