the position vectors of points x, y and z are x=ti +2j,y=2i+3j,z=3i+4j respectively. If the points x ,y and z are collinear, find the value of t
To check if the points x, y, and z are collinear, we need to determine if the vectors formed by these points are parallel.
The vector from x to y is given by y - x:
y - x = (2i + 3j) - (ti + 2j)
= (2 - t)i + (3 - 2)j
= (2 - t)i + j
The vector from y to z is given by z - y:
z - y = (3i + 4j) - (2i + 3j)
= (3 - 2)i + (4 - 3)j
= i + j
If x, y, and z are collinear, then the vectors (2 - t)i + j and i + j should be parallel. Two vectors are parallel if their components are proportional. In this case, the x-components and y-components of these vectors should be proportional.
Comparing the x-components:
2 - t = 1
Solving this equation for t gives:
t = 2 - 1
t = 1
Therefore, the value of t is 1.
To determine if the points x, y, and z are collinear, we need to check if the vectors from x to y and from x to z are parallel.
The vector from x to y is given by the difference between the position vectors of y and x:
xy = y - x = (2i + 3j) - (ti + 2j) = (2 - t)i + (3 - 2)j = (2 - t)i + j
The vector from x to z is given by the difference between the position vectors of z and x:
xz = z - x = (3i + 4j) - (ti + 2j) = (3 - t)i + (4 - 2)j = (3 - t)i + 2j
For these vectors to be parallel, their corresponding components must be proportional.
Comparing the i-components of xy and xz, we have:
2 - t = 3 - t
Simplifying the equation, we get:
2 = 3
Since the equation is not true for any value of t, the points x, y, and z are not collinear.
Therefore, there is no value of t that satisfies the condition for collinearity.