Please show ALL work Thank youSolutionSuppose that we are gi

Solution

Suppose that we are given two points on the line P₀ = (x_0, y_o, z₀) and P₁ = ( x₁, y₁, z₁ ).

Then P₀P₁ = < x₁ - x₀ , y₁ - y₀ , z₁ - z₀ > = < a,b,c> is a direction vector for the line.

Using P₀ for the point and P₀P₁ for the direction vector, we see that the parametric equations of the line are ;-

x = x₀ + t( x₁ - x₀ )

y = y₀ + t( y₁ -  y₀ )

z = z₀ + t( z₁ - z₀ )

This is usually written di§erently. If we distribute t and factor differently, we get

x = ( 1 - t ) x₀ + tx₁

y = ( 1 - t ) y₀ + ty₁

z = ( 1 - t ) z₀ + tz₁

When t = 0, we are at the point P₀ and when t = 1, we are at the point P₁. So, if t is allowed to take on any real value, then this equation will describe the whole line. On the other hand, if we restrict t to [0; 1], then this equation describes the portion of the line between P₀ and P₁ which is called the line segment from P₀ to P₁.

hence,putting the given value..

we get,

x = -( 1 - t )2 + 5t = ( t - 1 )*2 +5 t

y = ( 1 - t )*6 + 7t

z = ( 1- t )*1 - 8t = ( 1 - t ) - 8t

now,

The vector valued function-

Vector-Valued Function
   r(t) = (x₁ + at)i + (y₁ + bt)j + (z₁ + ct)k

= ( 5 + 7t )i + ( 7 + t )j + ( -8 - 9t )k