Test: Dimensional Geometry - 2 - Mathematics MCQ

Comments on the Area of a Δand a quadrilateral
The above formula for the area of a triangle or a quadrilateral and are easy to remember.
Write the coordinates as shown below:

Note that the first written coordinate is repeated in the last. Multiply diagonally along arrows. Attach a positive sign if the arrow goes from left to right and a negative sign if the arrow goes from right to left.

Proof: Let the  given points be

= Area of triangle ABC + Area of triangle BCD

OA is the initial line

The point (-4, -π/4) will lie in the 3rd quadrant and its coordinates in rectangular (cartesian) coordinates shall be

[nagative sing because P lies in the quadrant.]
∴ The polar coordinates of the point (3,-4) are

 The distance d between the points (2, 40°) and (4, 100°) is given by

Let P(r, θ) be the point whose distance from the point A(5, π/2) is 3.
∴PA = 3

This is the required  locus.

It si a simple matter to verify that

About Direction Cosines and Direction Ratios.
1. If a given line makes angles α, β, γ with positive directions of the axes of x, y and x respectively, then cosα, cosβ, cosγ are called the direction cosines (in short d.c.’s) of the given line.
Direction cosines of a line are generally denoted by l, m, n and are written as [l, m, n]

where r is the distance between the given points.
In particular, the d.c.'s of a line joining a point (a₁, y₁, z₁) to the origin are

4. Direction Cosines satisfy


Thus if the d.r.’s of a line are a,b, c, then the direction cosines of the line are

In fact, statement (a) is


Note: About Projections:
1. Projection of a point Pon a line is the foot of perpendicular from Pon the line.
2. Projection of a point P on a plane is the foot of perpendicular from Pon the plane.
3. Projection of a line segment PQ on another lin e A B is P' Q', where P' and Q' are respectively the projections of P and Q on the line AB. If θ is the angle between the lines PQ' and AB, then P' Q' - PQ cos θ 
4. Projection PQ' of a line joining two points P( x₁, y₁ z₁) and Q(x₂, y₂, z₂) on another line whose direction cosines are [l, m, n] is given by
P Q' - l{x₂ - x₁) + m(y₂ - y₁) + n(z₂ - z₁ )
5. Projection of a broken line consisting of continuous segments PQ₁, Q₁ Q₂, Q₂ Q₃ ..., Q_n Q on another line AB is the same as the projection oflinc PQ on AB.