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Determine whether the lines $ L_1 $ and $ L_2 $ are parallel, skew, or intersecting. If they intersect, find the point of intersection.

$ L_1 : x = 5 - 12t , y = 3 + 9t , z = 1 - 3t $$ L_2 : x = 3 + 8s , y = -6s , z = 7 + 2s $

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01:05

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 5

Equations of Lines and Planes

Vectors

Johns Hopkins University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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Determine whether the line…

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In the question they're asking to determine whether L- one and L- two are parallel skew intersecting and if they are intersecting them to find the point of intersection. Now in the question, it is given that line, L one has coordinates X equal to five minus 20. Why equal to three plus 90? That equal to one minus treaty? And L two line has coordinates X equal to three plus eight. S. Why? Equal to minus success? And they're equal to seven plus to us. So according to the question, you have to first check whether L one L two apparel it or not. So from that too. The questions of lying given there are coefficients of The three coordinates. If we find the ratio between the coordinates of both the lines then if these ratios are equal then these lines are parallel, otherwise they are not parallel. So the coefficients of L one In x coordinate is -12 and L two in x coordinate is less it. Similarly, everyone has Coefficient in Y. Coordinate as Last nine. And L two has coefficient in white board in it as -6 and L one has coefficient Z coordinate as minus three. And L two has coefficient in z coordinate as plus two. So if we compute the issue of both, the lines were efficient stones, then we get that it is equal to minus 12 by eight is equal to nine by minus six Is equal to -3 by two. And if we simplify this ratio we get It is equal to -3 x two for all the issues. And hence Mhm. These ratios are all equal and so here L. one and L two are parallel to each other and so we don't have to take the other criteria that whether it is Q. Or intersecting us, it needs to satisfy anyone of the condition That is here L one and L two are both parallel to each other, and so this is the required answer of the year in question.

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