RD Sharma Class 12 Chapter 27 Solutions (Direction Cosines And Direction Ratios)
RD Sharma Class 12 Maths Solutions Chapter 27 ‘Direction Cosines and Direction Ratios’ include various topics discussed in the chapter. Some of the main topics are recapitulation, coordinates of a point in space, signs of coordinates of a point, distance formula, section formulae, direction cosines and direction ratios of a line, angle between two vectors, the angle between two vectors in terms of their direction cosines, the angle between two vectors in terms of their direction ratios, algorithms, etc
RD Sharma Solutions Chapter 27 has 1 exercise of 16 questions where you will be asked about applying the concept of Direction Cosines and Direction Ratios. Though the chapter takes a small portion of your CBSE exams, yet it is very important to study it to get a full score in questions on this chapter. This chapter is a test of your understanding of all other chapters of Class 12 Maths and how you apply their concepts in Direction Cosines and Direction Ratios.
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Topics Discussed in RD Sharma Class 12 Maths Solutions Chapter 27 – Direction Cosines and Direction Ratios
- Coordinates of a point in space
When the three lines in space are mutually perpendicular to each other then they have planes that are mutually perpendicular to each other. This, in turn, divides the given space further into eight parts which we call octants. Call the lines coordinate axes.
Let the three mutually perpendicular lines are X’OX, Y’OY, and Z’OZ. The three lines meet at O. They meet in such a way that the two lines say Y’OY and Z’OZ are lying in the plane of the paper. The third line lies perpendicular to the paper’s plane and is projecting out of it.
X’OX is x-axis.
Y’OY is y-axis.
Z’OZ is z-axis.
O is the origin.
The three lines are known as coordinates’ rectangular axes. The planes that comprise lines X’OX, Y’OY, and Z’OZ in fixed pairs determine the three planes XOY, YOZ and ZOX that lie mutually perpendicular to each other. XY, YZ and ZX are known as rectangular coordinate planes.
- Signs of coordinates of a point
If you need to determine the coordinate signs of a given point in three dimensions, you will have to follow the sign convention which is analogous to the convention that you get in two-dimensional geometry. All distances measured along or that are parallel to OX, OY and OZ will always be positive. The distances measured parallel to OX’, OY’ and OZ’ will always be negative.
Now the octant with OX, OY and OZ as the edges can be easily denoted as OXYZ. And the other octants you get are OX’YZ, OXY’Z, OXYZ’, OX’Y’Z, OX’YZ’ OXY’Z’, and OX’Y’Z’.
The coordinates’ signs of a point are determined by the octant in which the octant lies. Consider P a point. Consider A, B, and C as the perpendicular’s feet drawn out of P point on X’OX, Y’OY, and Z’OZ respectively.
Discussion of Exercises in RD Sharma Class 12 Maths Solutions Chapter 27 Direction Cosines and Direction Ratios
The Chapter 27 Direction Cosines and Direction Ratios have questions such as finding the direction cosines where a given line makes certain angles with the axes, finding the acute angles between the two lines, showing the given points are collinear, showing the two lines are parallel, finding the angles between the two lines, given the direction cosines find the angle between the lines, finding the angle between the two vectors, proving the three points are collinear, etc.
Consider P(x1, y1, z1) and Q(x2, y2, z2) the two points. Consider R a point present on the line that joins both P and Q in such a way that it divides P and Q in the ratio m1m2.
Consider R coordinates as (((m1x2 + m2x1)/(m1+m2)), (m1y2+m2y1)/(m1+m2), (m1z2+m2z1)/m1+m2)))
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