ONE MORE STEP TOWARD BETTER TOMMOROW
Saturday, 9 August 2014
CHAPTER 3 - Worked Out Examples
How many lines can we draw that are equally inclined to each of the three coordinate axis?
Intuitively, we can expect the answer to be
, one for each of the
octants. Lets try to derive this answer rigorously.
Assume the direction cosines of the lines to be
But since the lines are equally inclined to the three axes, we have
This gives using
It is obvious that
are possible. Hence,
lines can be drawn which are equally inclined to the axes.
Find the direction cosines of the line segment joining
Refer to Fig –
. Note that the
-components of the segment
respectively. If the direction cosines of
and the length of
, we have
Thus, the direction cosines of
are given by
This result is quite important and will be used frequently in subsequent discussions.
Find the projection of the line segment joining the points
onto a line with direction cosines
Let us first consider a vector approach to this problem. The vector
can be written as
A unit vector
along the line with direction cosines
Therefore, the projected length of
upon this line will be.
This assertion can also be proved without resorting to the use of vectors. For this, we first understand the projection of a sequence of line segments on a given line.
points in space. The sum of projections of the sequence of segments
onto a fixed line
will be the same as the projection of
. This should be obvious from the following diagram:
The projection of the segment
The sum of projections of segments
We use this fact in our original problem as follows:
onto any line
(with direction cosines say
) will be sum of projections of
respectively, we get the total projection of
August 09, 2014
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