Planes & Vectors : Points and Equations of Parallel Lines

4) Line L is defined by the following equation:
Line M is defined by:
Find a and b, if known that lines are parallel to each other.

5) Line L is defined by intersection of two planes
2x+3y?z = 1 and -2x+y+2z = 0
Plane P is defined by -2x + y ?z = 4
Find any directional vector for the line, point of intersection of line and plane

6) Points A(0,1,3), B(-l, 3, 7), C( -4, 1, 0) and D(a, -1, 0). For which value of a will all four points lie in the same plane?

Please see the attached file for the fully formatted problems.

Please see the attached file for the complete solution.
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4. A standard form for the equations of a line is:

x - x0 y - y0 z - z0
------ = ------ = ------
a b c

Here (x0, y0, z0) is a point on the line, and the numbers a, b, and c determine the direction along the line: the vector a*I = b*J + c*K is parallel to the line.
For the first line, i = 3j - 5k, or v1 = i - 3j + 5k
For the second line, 3i = bj + ak, or v2 = 3i - bj - ak
If these two vectors are parallel to each other, then v1 x v2 = ...

Solution Summary

Points are found that allow lines to be parallel or in the same plane. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

You are given the vectors
X = (1,1,1), y = (2,1,1) and z = (6,2,2).
(i) Find the Cartesian equation of the plane Π normal to the vector x containing the point (2,1,1).
(ii) Find the parametric equation of the line l through the points (2,1,1) and (6,2,2).
(iii) If l' is given parametrically by l' = x + ty (with x

Please see attachment. Require problems solving, also explanations etc for better understanding of vectors.
VECTOR PROBLEMS
(1) Let l be the line with equation v = a + t u.
Show that the shortest distance from the origin to l can be written | a × u |

1.For the pairs of lines defined by the following equations indicate with an "I" if they are identical, a "P" if they are distinct but parallel, an "N" (for "normal") if they are perpendicular, and a "G" (for "general") if they are neither parallel nor perpendicular.
3x + 4y + 5 = 0 and y = - 3
4 x - 54 .
x = 2 and y = p

(1) a. Find the (vector) equation of the plane passing through the points (1,2,-2),
(-1,1,-9), (2,-2,-12).
b. Find the (vector) equation of the plane containing (1,2,-1) and perpendicular to
(3,-1,2).
(2) Suppose a, b, c are non zero vectors.
a. Explain why (a x b) x (a x c)

Please help with the following problems. Provide step by step calculations.
Find the slope of each line that has a slope
x=5y
Find an equation in the form y=ms +b (where possible) for each line.
Through (3,-5) parallel to y= 4
Graph each linear equation defined as follows;
3x - 5y =15
Solve each system of equat

WRITING AN EQUATION FOR THE LINE CONTAINING THE INDICATED POINTS:
1. (0,0) AND (3, 30)
2. (-4, -4) AND (-3, -3)
3. (-6, -6) AND (-3, 1)
4. (4, -8) AND (3, -6)
5. ( -1/2, 7) AND -4, 1/2)
6.(-9, 1) AND (-1/2, 1)
Those are the types of problems I am having trouble in writing equations containing indicated points. How do