Show that the points (2,3,4),(-1,-2,1),(5,8,7)are collinear.
Solution and Explanation
The given points are A(2,3,4),B(-1,-2,1),and C(5,8,7).
It is known that the direction ratio of line joining the points(x1,y1,z1),and(x2,y2,z2),
are given by,x2-x1,y2-y1,and z2-z1
The direction ratios of AB are(-1 -2),(-2-3),and(1-4)i.e.,-3,-5,and-3.
The direction ratios of BC are(5-(-1)),(8-(-2)),and(7,-1)i.e.,6,10,and 6.
It can be seen that the direction ratios of BC are -2 times that of AB i.e.,they are proportional.
Therefore,AB is parallel to BC.Since point B is common to both AB and BC,points A,B,and C are collinear.
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Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.