Projection In 3D Space

Dimension refers to the number of independent coordinates needed for defining the position of a point. Any real-life object can be referred to as a three-dimensional object. In three-dimensional space, there are three axes, named as x-axis, y-axis and z-axis. 

Let us consider there are two planes, plane-1 and plane-2. Now, plane-1 consists of some points on it. If we transform those points by connecting those points by parallel lines to plane-2, this transformation is known as projection.

 Direction Cosines of a Line

 Let us assume that the line OL made the angles α, β and with the positive direction of the x-axis, y-axis and z-axis, respectively (Fig. 1).

Fig. 1

Then, the cosine of those angles α, β, γ can be referred to as direction cosines of the given line. Direction cosines can be denoted as follows:

Let P be a point on the line OL and the distance OP=r.

Then, the position of the point P can be given by r , r , r or rl, rm, rn.

Representation of Three Direction Cosines

Consider a line RS, with the direction cosines l, m and n. A parallel line of RS is drawn, and P x, y, z is a point on the parallel line. Now, a perpendicular PA is drawn from the point P on the x-axis. (Fig. 2). Now, let us assume that distance OP =x2+y2+z2= r.

Thus, it can be written that x=rl, y=rm and z=rn.

Fig. 2

Now, in the right-angled OAP, =OAOP

  =xr

l=xr

x=lr

 Similarly, it can be shown that y = mr and z = nr.

Thus, x2+y2+z2=l2r2+m2r2+n2r2.

 It is also true that, x2+y2+z2 =r2.

Hence, r2= l2r2+m2r2+n2r2

l2+m2+n2=1.

 Direction Ratios of a Given Line

 It can be defined as any three numbers proportional to the direction cosines. 

Now, let us consider a line with direction cosines l, m and n and also assume that a, b and c are direction ratios.

 Thus, it can be written that a =λl, b=λm and c=λn, where is the proportionality constant. 

Relation Between Direction Ratios and Direction Cosines 

The direction cosines of a given line are l,m,n, and corresponding direction ratios are a,b,c. We know that direction cosines and corresponding direction ratios are proportional to each other. Thus, it can be written that

la=mb=nc=k

Thus, l=ak, m=bk and n=ck.

Since l2+m2+n2=1,

a2k2+b2k2+c2k2=1

k2 a2+b2+c2=1

k2=1a2+b2+c2

k= ±1a2+b2+c2

Thus,

l=ak=±aa2+b2+c2

m=bk=±ba2+b2+c2

n=ck=±ca2+b2+c2

Direction Cosines and Direction Ratios of a Line Passing Through Two Points

Let us consider a line that passes through two points P(x1, y1, z1) and Q(x2, y2, z2). The direction cosines are l,m,n. Perpendiculars PR and QS are drawn from P and Q on XY- plane. Now, another perpendicular PN is drawn from a point P on QS. (Fig. 3)

Fig. 3

Now, consider Fig. 4. In right-angled PNQ, cos =NQPQ=z2–z1PQ.

Fig. 4

Similarly, it can be shown that =x2–x1PQ and =y2–y1PQ.

Here, PQ denotes the distance between two points,

PQ=x2–x12+y2–y12+z2–z12 

The direction ratios of the line segment can be given by

a =x2–x1 or x1–x2

b =y2–y1 or y1–y2

c =z2–z1 or z1–z2

The Angle Between Two Lines

 Let L1 and L2 be two lines with direction cosines l1, m1, n1 and l2, m2, n2 respectively and direction ratios a1, b1, c1 and a2, b2, c2.

Now, if i, j and k are unit vectors along the axes. Then, the line can be represented in vector form by

L1=l1i+m1j+n1k

L2=l2i+m2j+n2k 

Let the angle between two lines be . Thus,

=L1⋅ L2L1L2

=l1i+m1j+n1kl2i+m2j+n2kl1i+m1j+n1kl2i+m2j+n2k

=l1l2+m1m2+n1n2l12+m12+n12l22+m22+n22                _ 1

 Now, we know that l2+m2+n2=1. Thus,

=l1l2+m1m2+n1n2           _ 2

We know that la=mb=nc=k. Thus, equation 1 can be written as

=a1a2+b1b2+c1c2k2k2a12+b12+c12a22+b22+c22

=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22                 _ 3