The direction ratios of the line which is perpendicular to the line

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Show that the points (4,7,8), (2,3,4), (-1,-2,1), (1,2,5) are vertices of a parallelogram.

1. Direction Cosines and Direction Ratios:

(i) Direction cosines of line:

If α, β, γ be the angles made by a line with x-axis, y-axis and z-axis respectively, then cosα, cosβ and cosγ are called direction cosines of a line, denoted by l, m and n respectively and the relation between l, m and n is given by l2+m2+n2=1. Direction cosine of x-axis, y-axis and z -axis are respectively 1, 0, 0, 0, 1, 0 and 0, 0, 1.

Important points:

(a) If a line makes α, β, γ with positive direction of x, y, z axes respectively, then direction cosines of line will be cosα, cosβ, cosγ or -cosα, -cosβ, -cosγ.

(b) sin2α+sin2β+ sin2 γ=2.

(ii) Direction ratios:

Any three numbers a, b, c proportional to direction cosines l, m, n are called direction ratios of the line., i.e., la=mb=nc. It is easy to see that there can be infinitely many sets of direction ratios for a given line.

(iii) Relation between D.C.'S and D.R.'S:

la=mb=nc

∴ l2a2=m2b2=n2c2=l2+m2+n2a2+b2+c2

∴ l=±aa2+b2+c2; m=±ba2+b2+c2; n=±ca2+b2+c2

(iv) Direction ratios and Direction cosines of the line joining two points:

Let Ax1, y1, z1 and Bx2, y2, z2 be two points, then d.r.'s of line AB are x2-x1, y2-y1, z2-z1 and the d.c.'s of AB are 1rx2-x1, 1ry2-y1, 1rz2-z1 where r=Σx2-x12|

2. Angle Between Two Intersecting Lines:

(i) Let θ be the acute angle between the lines with d.c.'s l1, m1, n1 and l2, m2, n2,

then cosθ=l1l2+m1m2+n1n2. If a1, b1, c1 and a2, b2, c2 be D.R.'s of two lines, then angle θ between them is given by cosθ=a1a2+b1b2+c1c2a12+b12+c12a22+b22+c22.

(ii) Perpendicularity and parallelism:

Let the two lines have their d.c.'s given by l1, m1, n1 and l2, m2, n2 respectively, then they are perpendicular if θ=90° i.e. cosθ=0, i.e. l1l2+m1m2+n1n2=0. Also the two lines are parallel if θ=0 i.e. sinθ=0, i.e. l1l2=m1m2=n1n2.

Note:

If instead of d.c.'s, d.r.'s a1, b1, c1 and a2, b2, c2 are given, then the lines are perpendicular if a1a2+b1b2+c1c2=0 and parallel if a1a2=b1b2=c1c2.

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