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?

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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.

... Q, R of BC, CA, AB have respective position vectors p = 1 ... A little vector algebra shows that t = m = n = 2/3 ... Indeed, AQ lies in a plane parallel to the xy-plane...

... expressible in terms of a fundamental vector operation, the ... of the two 5-d vectors (1,2,3 ... 1) definitions of such things as points, lines, planes, angles, circles ...

... Thus the vectors QP = 2i + 2j and RQ = 2i + j + 2k lie along the ... so d = 1 and the equation of the plane is. ... To find u, v and w, we need to find a vector V = ui ...

... 3 Let u , v and z are three vectors lie along the ... xa, (these lines are parallel to the miller indices plane in the ... dot product u, v and w with the vector g =ha ...

... a charge inside a dielectric; B for a point magnetic dipole ... an artist, I drew only two circles outside of the equatorial plane. ... c) Magnetic vector potential of ...

... and an additional charge cQ at point (0,0 ... thereof, such as cylindrical, spherical, elliptical, plane, etc ... for instance the field lines of the vector potential of ...

... The plane M' is defined by the vectors u and v, therefore ... denoted by the general vector Therefore an arbitrary vector that lies in the plane that contains ...

... At what angle above the horizontal is the plane's total displacement ... such problems is to write the vectors in component ... A vector can be resolved in two mutually ...

... c and the surface f x , y on the xy plane. ... The curl is a cross product between the vector field F ... A cross product between two parallel vectors is zero. ...

... of the wave (electromagnetic energy) is perpendicular to the plane in which the E and B vectors lie ...line integral of E, the electric field vector over any ...