Please show step by step solutions. Consider the points A(1,5, -1), B(3,2, -2) a
ID: 3286224 • Letter: P
Question
Please show step by step solutions.
Consider the points A(1,5, -1), B(3,2, -2) and C(1, 3,xi), where xi is a parameter (real number). Let P be the plane determined by the points A, B. and C. Let L be the line passing through A with the direction vector Find all values of the parameter xi so that the line L lies in the plane P. (Hint: Use the standard procedure to write the equation of P (a plane thorough three given points). This equation should depend on xi (alongside .x. y and z). Write down the parametric equations for L. Plug the parametric equations for L into the equation for P. This gives you an equation depending on t and xi. Now you need to impose a condition that will guarantee that the line L lies in the plane P. What values of xi will guarantee the fulfillment of the needed condition? Think about it. You observed this phenomenon in one of the numerical examples in class notes.)Explanation / Answer
The equation of the plane that is determined by the three points is of the form a(x-1)+b(y-5)+c(z+1)=0 such that they satisfy 2a-3b-c=0 and -2b+(@+1)c=0 @ is the symbol for epsilon the equation is solved to be (3@+5)x+2@+2)y+z=14+13@ since L passes through A, we need to only check that it is parallel to the plane because the distance of the line from the plane is zero so the normal of the plane should be perpendicular to the line direction ratios of the line are (1,-1,3), direction ratios of the normal to the plane are (3@+5,2@+2,1), if they are perpendicular, dot product is 0, implies (@+6)=0 so the value of epsilon should be (-6)